Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A

Percentage Accurate: 99.4% → 99.6%

Time: 6.5s

Alternatives: 23

Speedup: 1.0×

• 43.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * exp(((t * t) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.exp(((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.exp(((t * t) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * exp(Float64(Float64(t * t) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * exp(((t * t) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.exp(((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.exp(((t * t) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * exp(Float64(Float64(t * t) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\sqrt{z} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (* (sqrt z) (sqrt (pow (exp t) t))) (fma 0.5 x (- y))) (sqrt 2.0)))

C

double code(double x, double y, double z, double t) {
	return ((sqrt(z) * sqrt(pow(exp(t), t))) * fma(0.5, x, -y)) * sqrt(2.0);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(sqrt(z) * sqrt((exp(t) ^ t))) * fma(0.5, x, Float64(-y))) * sqrt(2.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[N[Power[N[Exp[t], $MachinePrecision], t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   3. lift--.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \color{blue}{\sqrt{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   8. sqrt-prod N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)} \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   10. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   11. *-commutative N/A

       \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   12. lift-exp.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{\frac{t \cdot t}{2}}} \]

   13. lift-*.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{t \cdot t}}{2}} \]

   14. lift-/.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{t \cdot t}{2}}} \]

   15. pow2 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{{t}^{2}}}{2}} \]

   16. exp-sqrt N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

   17. pow1/2 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{{\left(e^{{t}^{2}}\right)}^{\frac{1}{2}}} \]

   18. exp-prod N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{{t}^{2} \cdot \frac{1}{2}}} \]

   19. *-commutative N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{1}{2} \cdot {t}^{2}}} \]

3. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\left(\sqrt{z} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2}} \]
4. Add Preprocessing

Alternative 2: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(x, 0.5, -y\right) \cdot \sqrt{z \cdot {\left(1 + t\right)}^{t}}\right) \cdot \sqrt{2} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (fma x 0.5 (- y)) (sqrt (* z (pow (+ 1.0 t) t)))) (sqrt 2.0)))

C

double code(double x, double y, double z, double t) {
	return (fma(x, 0.5, -y) * sqrt((z * pow((1.0 + t), t)))) * sqrt(2.0);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(fma(x, 0.5, Float64(-y)) * sqrt(Float64(z * (Float64(1.0 + t) ^ t)))) * sqrt(2.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5 + (-y)), $MachinePrecision] * N[Sqrt[N[(z * N[Power[N[(1.0 + t), $MachinePrecision], t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   3. lift--.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \color{blue}{\sqrt{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   8. sqrt-prod N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)} \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   10. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   11. *-commutative N/A

       \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   12. lift-exp.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{\frac{t \cdot t}{2}}} \]

   13. lift-*.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{t \cdot t}}{2}} \]

   14. lift-/.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{t \cdot t}{2}}} \]

   15. pow2 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{{t}^{2}}}{2}} \]

   16. exp-sqrt N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

   17. pow1/2 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{{\left(e^{{t}^{2}}\right)}^{\frac{1}{2}}} \]

   18. exp-prod N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{{t}^{2} \cdot \frac{1}{2}}} \]

   19. *-commutative N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{1}{2} \cdot {t}^{2}}} \]

3. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\left(\sqrt{z} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2}} \]

4. Taylor expanded in t around 0

   \[\leadsto \left(\left(\sqrt{z} \cdot \sqrt{{\color{blue}{\left(1 + t\right)}}^{t}}\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]
5. Step-by-step derivation

   1. lower-+.f64 75.4

      \[\leadsto \left(\left(\sqrt{z} \cdot \sqrt{{\left(1 + \color{blue}{t}\right)}^{t}}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]

6. Applied rewrites 75.4%

   \[\leadsto \left(\left(\sqrt{z} \cdot \sqrt{{\color{blue}{\left(1 + t\right)}}^{t}}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{z} \cdot \sqrt{{\left(1 + t\right)}^{t}}\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right)} \cdot \sqrt{2} \]

   2. lift-fma.f64 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \sqrt{{\left(1 + t\right)}^{t}}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x + \left(-y\right)\right)}\right) \cdot \sqrt{2} \]

   3. lift-neg.f64 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \sqrt{{\left(1 + t\right)}^{t}}\right) \cdot \left(\frac{1}{2} \cdot x + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \cdot \sqrt{2} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(y\right)\right)\right) \cdot \left(\sqrt{z} \cdot \sqrt{{\left(1 + t\right)}^{t}}\right)\right)} \cdot \sqrt{2} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(y\right)\right)\right) \cdot \left(\sqrt{z} \cdot \sqrt{{\left(1 + t\right)}^{t}}\right)\right)} \cdot \sqrt{2} \]

   6. *-commutative N/A

      \[\leadsto \left(\left(\color{blue}{x \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(y\right)\right)\right) \cdot \left(\sqrt{z} \cdot \sqrt{{\left(1 + t\right)}^{t}}\right)\right) \cdot \sqrt{2} \]

   7. lift-neg.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} + \color{blue}{\left(-y\right)}\right) \cdot \left(\sqrt{z} \cdot \sqrt{{\left(1 + t\right)}^{t}}\right)\right) \cdot \sqrt{2} \]

   8. lower-fma.f64 75.4

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(x, 0.5, -y\right)} \cdot \left(\sqrt{z} \cdot \sqrt{{\left(1 + t\right)}^{t}}\right)\right) \cdot \sqrt{2} \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(x, \frac{1}{2}, -y\right) \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{{\left(1 + t\right)}^{t}}\right)}\right) \cdot \sqrt{2} \]

   10. lift-sqrt.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x, \frac{1}{2}, -y\right) \cdot \left(\color{blue}{\sqrt{z}} \cdot \sqrt{{\left(1 + t\right)}^{t}}\right)\right) \cdot \sqrt{2} \]

   11. lift-sqrt.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x, \frac{1}{2}, -y\right) \cdot \left(\sqrt{z} \cdot \color{blue}{\sqrt{{\left(1 + t\right)}^{t}}}\right)\right) \cdot \sqrt{2} \]

   12. sqrt-unprod N/A

       \[\leadsto \left(\mathsf{fma}\left(x, \frac{1}{2}, -y\right) \cdot \color{blue}{\sqrt{z \cdot {\left(1 + t\right)}^{t}}}\right) \cdot \sqrt{2} \]

   13. lower-sqrt.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x, \frac{1}{2}, -y\right) \cdot \color{blue}{\sqrt{z \cdot {\left(1 + t\right)}^{t}}}\right) \cdot \sqrt{2} \]

   14. lower-*.f64 75.4

       \[\leadsto \left(\mathsf{fma}\left(x, 0.5, -y\right) \cdot \sqrt{\color{blue}{z \cdot {\left(1 + t\right)}^{t}}}\right) \cdot \sqrt{2} \]

8. Applied rewrites 75.4%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, 0.5, -y\right) \cdot \sqrt{z \cdot {\left(1 + t\right)}^{t}}\right) \cdot \sqrt{2}} \]
9. Add Preprocessing

Alternative 3: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\left(t \cdot t\right) \cdot 0.5} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (* (* t t) 0.5))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) * 0.5));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * exp(((t * t) * 0.5d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.exp(((t * t) * 0.5));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.exp(((t * t) * 0.5))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * exp(Float64(Float64(t * t) * 0.5)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) * 0.5));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{e^{\frac{t \cdot t}{2}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{\color{blue}{t \cdot t}}{2}} \]

   3. lift-/.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{\frac{t \cdot t}{2}}} \]

   4. pow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{\color{blue}{{t}^{2}}}{2}} \]

   5. exp-sqrt N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

   6. pow1/2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{{\left(e^{{t}^{2}}\right)}^{\frac{1}{2}}} \]

   7. exp-prod N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{e^{{t}^{2} \cdot \frac{1}{2}}} \]

   8. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{\frac{1}{2} \cdot {t}^{2}}} \]

   9. lower-exp.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{e^{\frac{1}{2} \cdot {t}^{2}}} \]

   10. *-commutative N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{{t}^{2} \cdot \frac{1}{2}}} \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{{t}^{2} \cdot \frac{1}{2}}} \]

   12. pow2 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{\left(t \cdot t\right)} \cdot \frac{1}{2}} \]

   13. lift-*.f64 99.4

       \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{\left(t \cdot t\right)} \cdot 0.5} \]

3. Applied rewrites 99.4%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{e^{\left(t \cdot t\right) \cdot 0.5}} \]
4. Add Preprocessing

Alternative 4: 95.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right)\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (*
   (*
    (sqrt z)
    (fma
     (fma (fma 0.020833333333333332 (* t t) 0.125) (* t t) 0.5)
     (* t t)
     1.0))
   (fma 0.5 x (- y)))
  (sqrt 2.0)))

C

double code(double x, double y, double z, double t) {
	return ((sqrt(z) * fma(fma(fma(0.020833333333333332, (t * t), 0.125), (t * t), 0.5), (t * t), 1.0)) * fma(0.5, x, -y)) * sqrt(2.0);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(sqrt(z) * fma(fma(fma(0.020833333333333332, Float64(t * t), 0.125), Float64(t * t), 0.5), Float64(t * t), 1.0)) * fma(0.5, x, Float64(-y))) * sqrt(2.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[Sqrt[z], $MachinePrecision] * N[(N[(N[(0.020833333333333332 * N[(t * t), $MachinePrecision] + 0.125), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   3. lift--.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \color{blue}{\sqrt{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   8. sqrt-prod N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)} \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   10. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   11. *-commutative N/A

       \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   12. lift-exp.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{\frac{t \cdot t}{2}}} \]

   13. lift-*.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{t \cdot t}}{2}} \]

   14. lift-/.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{t \cdot t}{2}}} \]

   15. pow2 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{{t}^{2}}}{2}} \]

   16. exp-sqrt N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

   17. pow1/2 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{{\left(e^{{t}^{2}}\right)}^{\frac{1}{2}}} \]

   18. exp-prod N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{{t}^{2} \cdot \frac{1}{2}}} \]

   19. *-commutative N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{1}{2} \cdot {t}^{2}}} \]

3. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\left(\sqrt{z} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2}} \]

4. Taylor expanded in t around 0

   \[\leadsto \left(\left(\sqrt{z} \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)}\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \left({t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) + \color{blue}{1}\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   2. *-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \left(\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) \cdot {t}^{2} + 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right), \color{blue}{{t}^{2}}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   4. +-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left({t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) + \frac{1}{2}, {\color{blue}{t}}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   5. *-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) \cdot {t}^{2} + \frac{1}{2}, {t}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}, {t}^{2}, \frac{1}{2}\right), {\color{blue}{t}}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   7. +-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48} \cdot {t}^{2} + \frac{1}{8}, {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, {t}^{2}, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   9. pow2 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   11. pow2 N/A

       \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   12. lift-*.f64 N/A

       \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   13. pow2 N/A

       \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), t \cdot \color{blue}{t}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   14. lift-*.f64 95.0

       \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot \color{blue}{t}, 1\right)\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]

6. Applied rewrites 95.0%

   \[\leadsto \left(\left(\sqrt{z} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right)}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]
7. Add Preprocessing

Alternative 5: 95.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right) \cdot \mathsf{fma}\left(x, 0.5, -y\right)\right)\right) \cdot \sqrt{2} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (*
   (sqrt z)
   (*
    (fma
     (fma (fma 0.020833333333333332 (* t t) 0.125) (* t t) 0.5)
     (* t t)
     1.0)
    (fma x 0.5 (- y))))
  (sqrt 2.0)))

C

double code(double x, double y, double z, double t) {
	return (sqrt(z) * (fma(fma(fma(0.020833333333333332, (t * t), 0.125), (t * t), 0.5), (t * t), 1.0) * fma(x, 0.5, -y))) * sqrt(2.0);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(sqrt(z) * Float64(fma(fma(fma(0.020833333333333332, Float64(t * t), 0.125), Float64(t * t), 0.5), Float64(t * t), 1.0) * fma(x, 0.5, Float64(-y)))) * sqrt(2.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[Sqrt[z], $MachinePrecision] * N[(N[(N[(N[(0.020833333333333332 * N[(t * t), $MachinePrecision] + 0.125), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * 0.5 + (-y)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   3. lift--.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \color{blue}{\sqrt{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   8. sqrt-prod N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)} \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   10. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   11. *-commutative N/A

       \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   12. lift-exp.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{\frac{t \cdot t}{2}}} \]

   13. lift-*.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{t \cdot t}}{2}} \]

   14. lift-/.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{t \cdot t}{2}}} \]

   15. pow2 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{{t}^{2}}}{2}} \]

   16. exp-sqrt N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

   17. pow1/2 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{{\left(e^{{t}^{2}}\right)}^{\frac{1}{2}}} \]

   18. exp-prod N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{{t}^{2} \cdot \frac{1}{2}}} \]

   19. *-commutative N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{1}{2} \cdot {t}^{2}}} \]

3. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\left(\sqrt{z} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2}} \]

4. Taylor expanded in t around 0

   \[\leadsto \left(\left(\sqrt{z} \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \left(\frac{1}{2} \cdot {t}^{2} + \color{blue}{1}\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   2. *-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \left({t}^{2} \cdot \frac{1}{2} + 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left({t}^{2}, \color{blue}{\frac{1}{2}}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   4. pow2 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   5. lift-*.f64 86.1

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]

6. Applied rewrites 86.1%

   \[\leadsto \left(\left(\sqrt{z} \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, 0.5, 1\right)}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right)} \cdot \sqrt{2} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right)} \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\sqrt{z}} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   4. lift-fma.f64 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x + \left(-y\right)\right)}\right) \cdot \sqrt{2} \]

   5. lift-neg.f64 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \cdot \left(\frac{1}{2} \cdot x + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \cdot \sqrt{2} \]

   6. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)} \cdot \sqrt{2} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)} \cdot \sqrt{2} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \left(\color{blue}{\sqrt{z}} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right) \cdot \sqrt{2} \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\sqrt{z} \cdot \color{blue}{\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}\right) \cdot \sqrt{2} \]

   10. *-commutative N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(\color{blue}{x \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right) \cdot \sqrt{2} \]

   11. lift-neg.f64 N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(x \cdot \frac{1}{2} + \color{blue}{\left(-y\right)}\right)\right)\right) \cdot \sqrt{2} \]

   12. lower-fma.f64 87.6

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(x, 0.5, -y\right)}\right)\right) \cdot \sqrt{2} \]

8. Applied rewrites 87.6%

   \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right) \cdot \mathsf{fma}\left(x, 0.5, -y\right)\right)\right)} \cdot \sqrt{2} \]

9. Taylor expanded in t around 0

   \[\leadsto \left(\sqrt{z} \cdot \left(\color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)} \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\left({t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) + \color{blue}{1}\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

    2. *-commutative N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\left(\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) \cdot {t}^{2} + 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

    3. lower-fma.f64 N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right), \color{blue}{{t}^{2}}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

    4. +-commutative N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left({t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) + \frac{1}{2}, {\color{blue}{t}}^{2}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

    5. *-commutative N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) \cdot {t}^{2} + \frac{1}{2}, {t}^{2}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

    6. lower-fma.f64 N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}, {t}^{2}, \frac{1}{2}\right), {\color{blue}{t}}^{2}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

    7. +-commutative N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48} \cdot {t}^{2} + \frac{1}{8}, {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

    8. lower-fma.f64 N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, {t}^{2}, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

    9. pow2 N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

    10. lift-*.f64 N/A

        \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

    11. pow2 N/A

        \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

    12. lift-*.f64 N/A

        \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

    13. pow2 N/A

        \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), t \cdot \color{blue}{t}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

    14. lift-*.f64 95.4

        \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot \color{blue}{t}, 1\right) \cdot \mathsf{fma}\left(x, 0.5, -y\right)\right)\right) \cdot \sqrt{2} \]

11. Applied rewrites 95.4%

    \[\leadsto \left(\sqrt{z} \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right)} \cdot \mathsf{fma}\left(x, 0.5, -y\right)\right)\right) \cdot \sqrt{2} \]
12. Add Preprocessing

Alternative 6: 94.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (* (- (* x 0.5) y) (sqrt (* z 2.0)))
  (fma
   (fma (fma 0.020833333333333332 (* t t) 0.125) (* t t) 0.5)
   (* t t)
   1.0)))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * fma(fma(fma(0.020833333333333332, (t * t), 0.125), (t * t), 0.5), (t * t), 1.0);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * fma(fma(fma(0.020833333333333332, Float64(t * t), 0.125), Float64(t * t), 0.5), Float64(t * t), 1.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.020833333333333332 * N[(t * t), $MachinePrecision] + 0.125), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

2. Taylor expanded in t around 0

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) + \color{blue}{1}\right) \]

   2. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) \cdot {t}^{2} + 1\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right), \color{blue}{{t}^{2}}, 1\right) \]

   4. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left({t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) + \frac{1}{2}, {\color{blue}{t}}^{2}, 1\right) \]

   5. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) \cdot {t}^{2} + \frac{1}{2}, {t}^{2}, 1\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}, {t}^{2}, \frac{1}{2}\right), {\color{blue}{t}}^{2}, 1\right) \]

   7. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48} \cdot {t}^{2} + \frac{1}{8}, {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, {t}^{2}, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   9. pow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   11. pow2 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   12. lift-*.f64 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   13. pow2 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), t \cdot \color{blue}{t}, 1\right) \]

   14. lift-*.f64 94.5

       \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot \color{blue}{t}, 1\right) \]

4. Applied rewrites 94.5%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right)} \]
5. Add Preprocessing

Alternative 7: 72.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \sqrt{z}, \left(-y\right) \cdot \sqrt{z}\right) \cdot \sqrt{2}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{+115}:\\ \;\;\;\;\left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right), t \cdot t, 1\right)} \cdot 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot 0.5\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 7.8e+43)
   (* (fma (* 0.5 x) (sqrt z) (* (- y) (sqrt z))) (sqrt 2.0))
   (if (<= t 3.9e+115)
     (* (* (sqrt (* (* 2.0 z) (fma (fma (* t t) 0.5 1.0) (* t t) 1.0))) 0.5) x)
     (* (sqrt z) (* (* (* t t) 0.5) (* (- (* x 0.5) y) (sqrt 2.0)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 7.8e+43) {
		tmp = fma((0.5 * x), sqrt(z), (-y * sqrt(z))) * sqrt(2.0);
	} else if (t <= 3.9e+115) {
		tmp = (sqrt(((2.0 * z) * fma(fma((t * t), 0.5, 1.0), (t * t), 1.0))) * 0.5) * x;
	} else {
		tmp = sqrt(z) * (((t * t) * 0.5) * (((x * 0.5) - y) * sqrt(2.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 7.8e+43)
		tmp = Float64(fma(Float64(0.5 * x), sqrt(z), Float64(Float64(-y) * sqrt(z))) * sqrt(2.0));
	elseif (t <= 3.9e+115)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * z) * fma(fma(Float64(t * t), 0.5, 1.0), Float64(t * t), 1.0))) * 0.5) * x);
	else
		tmp = Float64(sqrt(z) * Float64(Float64(Float64(t * t) * 0.5) * Float64(Float64(Float64(x * 0.5) - y) * sqrt(2.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 7.8e+43], N[(N[(N[(0.5 * x), $MachinePrecision] * N[Sqrt[z], $MachinePrecision] + N[((-y) * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.9e+115], N[(N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision], N[(N[Sqrt[z], $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.8000000000000001e43

   1. Initial program 99.4%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \color{blue}{\sqrt{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      8. sqrt-prod N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)} \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      12. lift-exp.f64 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{\frac{t \cdot t}{2}}} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{t \cdot t}}{2}} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{t \cdot t}{2}}} \]

      15. pow2 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{{t}^{2}}}{2}} \]

      16. exp-sqrt N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

      17. pow1/2 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{{\left(e^{{t}^{2}}\right)}^{\frac{1}{2}}} \]

      18. exp-prod N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{{t}^{2} \cdot \frac{1}{2}}} \]

      19. *-commutative N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{1}{2} \cdot {t}^{2}}} \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2}} \]

   4. Taylor expanded in t around 0

      \[\leadsto \left(\color{blue}{\sqrt{z}} \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 68.9

         \[\leadsto \left(\sqrt{z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]

   6. Applied rewrites 68.9%

      \[\leadsto \left(\color{blue}{\sqrt{z}} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right)} \cdot \sqrt{2} \]

      2. lift-fma.f64 N/A

         \[\leadsto \left(\sqrt{z} \cdot \color{blue}{\left(\frac{1}{2} \cdot x + \left(-y\right)\right)}\right) \cdot \sqrt{2} \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot x\right) \cdot \sqrt{z} + \left(-y\right) \cdot \sqrt{z}\right)} \cdot \sqrt{2} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot x, \sqrt{z}, \left(-y\right) \cdot \sqrt{z}\right)} \cdot \sqrt{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot x}, \sqrt{z}, \left(-y\right) \cdot \sqrt{z}\right) \cdot \sqrt{2} \]

      6. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot x, \sqrt{\color{blue}{z}}, \left(-y\right) \cdot \sqrt{z}\right) \cdot \sqrt{2} \]

      7. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot x, \sqrt{\color{blue}{z}}, \mathsf{Rewrite=>}\left(lower-*.f64, \left(\left(-y\right) \cdot \sqrt{z}\right)\right)\right) \cdot \sqrt{2} \]

      8. sqrt-prod N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot x, \sqrt{\color{blue}{z}}, \left(-y\right) \cdot \sqrt{z}\right) \cdot \sqrt{2} \]

   8. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot x, \sqrt{z}, \left(-y\right) \cdot \sqrt{z}\right)} \cdot \sqrt{2} \]

   if 7.8000000000000001e43 < t < 3.90000000000000006e115

   1. Initial program 99.1%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \color{blue}{\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(\left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right) \cdot \color{blue}{x}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{x}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 75.7%

      \[\leadsto \color{blue}{\left(\sqrt{\left(2 \cdot z\right) \cdot {\left(e^{t}\right)}^{t}} \cdot 0.5\right) \cdot x} \]

   5. Taylor expanded in t around 0

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(1 + {t}^{2} \cdot \left(1 + \frac{1}{2} \cdot {t}^{2}\right)\right)} \cdot \frac{1}{2}\right) \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left({t}^{2} \cdot \left(1 + \frac{1}{2} \cdot {t}^{2}\right) + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(\left(1 + \frac{1}{2} \cdot {t}^{2}\right) \cdot {t}^{2} + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(1 + \frac{1}{2} \cdot {t}^{2}, {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      4. +-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\frac{1}{2} \cdot {t}^{2} + 1, {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left({t}^{2} \cdot \frac{1}{2} + 1, {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({t}^{2}, \frac{1}{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      7. pow2 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      9. pow2 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right), t \cdot t, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      10. lift-*.f64 56.5

          \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right), t \cdot t, 1\right)} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 56.5%

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right), t \cdot t, 1\right)} \cdot 0.5\right) \cdot x \]

   if 3.90000000000000006e115 < t

   1. Initial program 99.1%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) \cdot \sqrt{z}\right) + \sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)\right) \cdot \sqrt{z} + \color{blue}{\sqrt{z}} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)\right) \cdot \sqrt{z} + \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \color{blue}{\sqrt{z}} \]

      3. distribute-rgt-out N/A

         \[\leadsto \sqrt{z} \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) + \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{z} \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) + \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)} + \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) + \color{blue}{\sqrt{2}} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{z} \cdot \mathsf{fma}\left(\frac{1}{2} \cdot {t}^{2}, \color{blue}{\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)}, \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

   4. Applied rewrites 95.5%

      \[\leadsto \color{blue}{\sqrt{z} \cdot \mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.5, \sqrt{2} \cdot \mathsf{fma}\left(0.5, x, -y\right), \sqrt{2} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \sqrt{z} \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot x - y\right)}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot x - y\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{z} \cdot \left(\left({t}^{2} \cdot \frac{1}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right)\right)\right) \]

      4. pow2 N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot x - y\right)\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right)\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot x - y\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{1}{2} \cdot x - y\right) \cdot \sqrt{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{1}{2} \cdot x - y\right) \cdot \sqrt{2}\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{1}{2} \cdot x - y\right) \cdot \sqrt{2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{2}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{2}\right)\right) \]

      12. lift-sqrt.f64 95.5

          \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot 0.5\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)\right) \]

   7. Applied rewrites 95.5%

      \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot 0.5\right) \cdot \color{blue}{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 72.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{+43}:\\ \;\;\;\;\left(\sqrt{z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{+115}:\\ \;\;\;\;\left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right), t \cdot t, 1\right)} \cdot 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot 0.5\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 7.8e+43)
   (* (* (sqrt z) (fma 0.5 x (- y))) (sqrt 2.0))
   (if (<= t 3.9e+115)
     (* (* (sqrt (* (* 2.0 z) (fma (fma (* t t) 0.5 1.0) (* t t) 1.0))) 0.5) x)
     (* (sqrt z) (* (* (* t t) 0.5) (* (- (* x 0.5) y) (sqrt 2.0)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 7.8e+43) {
		tmp = (sqrt(z) * fma(0.5, x, -y)) * sqrt(2.0);
	} else if (t <= 3.9e+115) {
		tmp = (sqrt(((2.0 * z) * fma(fma((t * t), 0.5, 1.0), (t * t), 1.0))) * 0.5) * x;
	} else {
		tmp = sqrt(z) * (((t * t) * 0.5) * (((x * 0.5) - y) * sqrt(2.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 7.8e+43)
		tmp = Float64(Float64(sqrt(z) * fma(0.5, x, Float64(-y))) * sqrt(2.0));
	elseif (t <= 3.9e+115)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * z) * fma(fma(Float64(t * t), 0.5, 1.0), Float64(t * t), 1.0))) * 0.5) * x);
	else
		tmp = Float64(sqrt(z) * Float64(Float64(Float64(t * t) * 0.5) * Float64(Float64(Float64(x * 0.5) - y) * sqrt(2.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 7.8e+43], N[(N[(N[Sqrt[z], $MachinePrecision] * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.9e+115], N[(N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision], N[(N[Sqrt[z], $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.8000000000000001e43

   1. Initial program 99.4%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \color{blue}{\sqrt{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      8. sqrt-prod N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)} \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      12. lift-exp.f64 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{\frac{t \cdot t}{2}}} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{t \cdot t}}{2}} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{t \cdot t}{2}}} \]

      15. pow2 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{{t}^{2}}}{2}} \]

      16. exp-sqrt N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

      17. pow1/2 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{{\left(e^{{t}^{2}}\right)}^{\frac{1}{2}}} \]

      18. exp-prod N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{{t}^{2} \cdot \frac{1}{2}}} \]

      19. *-commutative N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{1}{2} \cdot {t}^{2}}} \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2}} \]

   4. Taylor expanded in t around 0

      \[\leadsto \left(\color{blue}{\sqrt{z}} \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 68.9

         \[\leadsto \left(\sqrt{z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]

   6. Applied rewrites 68.9%

      \[\leadsto \left(\color{blue}{\sqrt{z}} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]

   if 7.8000000000000001e43 < t < 3.90000000000000006e115

   1. Initial program 99.1%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \color{blue}{\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(\left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right) \cdot \color{blue}{x}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{x}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 75.7%

      \[\leadsto \color{blue}{\left(\sqrt{\left(2 \cdot z\right) \cdot {\left(e^{t}\right)}^{t}} \cdot 0.5\right) \cdot x} \]

   5. Taylor expanded in t around 0

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(1 + {t}^{2} \cdot \left(1 + \frac{1}{2} \cdot {t}^{2}\right)\right)} \cdot \frac{1}{2}\right) \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left({t}^{2} \cdot \left(1 + \frac{1}{2} \cdot {t}^{2}\right) + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(\left(1 + \frac{1}{2} \cdot {t}^{2}\right) \cdot {t}^{2} + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(1 + \frac{1}{2} \cdot {t}^{2}, {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      4. +-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\frac{1}{2} \cdot {t}^{2} + 1, {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left({t}^{2} \cdot \frac{1}{2} + 1, {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({t}^{2}, \frac{1}{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      7. pow2 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      9. pow2 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right), t \cdot t, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      10. lift-*.f64 56.5

          \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right), t \cdot t, 1\right)} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 56.5%

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right), t \cdot t, 1\right)} \cdot 0.5\right) \cdot x \]

   if 3.90000000000000006e115 < t

   1. Initial program 99.1%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) \cdot \sqrt{z}\right) + \sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)\right) \cdot \sqrt{z} + \color{blue}{\sqrt{z}} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)\right) \cdot \sqrt{z} + \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \color{blue}{\sqrt{z}} \]

      3. distribute-rgt-out N/A

         \[\leadsto \sqrt{z} \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) + \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{z} \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) + \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)} + \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) + \color{blue}{\sqrt{2}} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{z} \cdot \mathsf{fma}\left(\frac{1}{2} \cdot {t}^{2}, \color{blue}{\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)}, \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

   4. Applied rewrites 95.5%

      \[\leadsto \color{blue}{\sqrt{z} \cdot \mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.5, \sqrt{2} \cdot \mathsf{fma}\left(0.5, x, -y\right), \sqrt{2} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \sqrt{z} \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot x - y\right)}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot x - y\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{z} \cdot \left(\left({t}^{2} \cdot \frac{1}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right)\right)\right) \]

      4. pow2 N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot x - y\right)\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right)\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot x - y\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{1}{2} \cdot x - y\right) \cdot \sqrt{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{1}{2} \cdot x - y\right) \cdot \sqrt{2}\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{1}{2} \cdot x - y\right) \cdot \sqrt{2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{2}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{2}\right)\right) \]

      12. lift-sqrt.f64 95.5

          \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot 0.5\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)\right) \]

   7. Applied rewrites 95.5%

      \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot 0.5\right) \cdot \color{blue}{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 92.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.125, t \cdot t, 0.5\right), t \cdot t, 1\right)\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (* (* (sqrt z) (fma (fma 0.125 (* t t) 0.5) (* t t) 1.0)) (fma 0.5 x (- y)))
  (sqrt 2.0)))

C

double code(double x, double y, double z, double t) {
	return ((sqrt(z) * fma(fma(0.125, (t * t), 0.5), (t * t), 1.0)) * fma(0.5, x, -y)) * sqrt(2.0);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(sqrt(z) * fma(fma(0.125, Float64(t * t), 0.5), Float64(t * t), 1.0)) * fma(0.5, x, Float64(-y))) * sqrt(2.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[Sqrt[z], $MachinePrecision] * N[(N[(0.125 * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   3. lift--.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \color{blue}{\sqrt{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   8. sqrt-prod N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)} \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   10. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   11. *-commutative N/A

       \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   12. lift-exp.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{\frac{t \cdot t}{2}}} \]

   13. lift-*.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{t \cdot t}}{2}} \]

   14. lift-/.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{t \cdot t}{2}}} \]

   15. pow2 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{{t}^{2}}}{2}} \]

   16. exp-sqrt N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

   17. pow1/2 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{{\left(e^{{t}^{2}}\right)}^{\frac{1}{2}}} \]

   18. exp-prod N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{{t}^{2} \cdot \frac{1}{2}}} \]

   19. *-commutative N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{1}{2} \cdot {t}^{2}}} \]

3. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\left(\sqrt{z} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2}} \]

4. Taylor expanded in t around 0

   \[\leadsto \left(\left(\sqrt{z} \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right)\right)}\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \left({t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right) + \color{blue}{1}\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   2. *-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \left(\left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right) \cdot {t}^{2} + 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}, \color{blue}{{t}^{2}}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   4. +-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(\frac{1}{8} \cdot {t}^{2} + \frac{1}{2}, {\color{blue}{t}}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, {t}^{2}, \frac{1}{2}\right), {\color{blue}{t}}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   6. pow2 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   8. pow2 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, t \cdot t, \frac{1}{2}\right), t \cdot \color{blue}{t}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   9. lift-*.f64 92.7

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.125, t \cdot t, 0.5\right), t \cdot \color{blue}{t}, 1\right)\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]

6. Applied rewrites 92.7%

   \[\leadsto \left(\left(\sqrt{z} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.125, t \cdot t, 0.5\right), t \cdot t, 1\right)}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]
7. Add Preprocessing

Alternative 10: 93.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.125, t \cdot t, 0.5\right), t \cdot t, 1\right) \cdot \mathsf{fma}\left(x, 0.5, -y\right)\right)\right) \cdot \sqrt{2} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (* (sqrt z) (* (fma (fma 0.125 (* t t) 0.5) (* t t) 1.0) (fma x 0.5 (- y))))
  (sqrt 2.0)))

C

double code(double x, double y, double z, double t) {
	return (sqrt(z) * (fma(fma(0.125, (t * t), 0.5), (t * t), 1.0) * fma(x, 0.5, -y))) * sqrt(2.0);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(sqrt(z) * Float64(fma(fma(0.125, Float64(t * t), 0.5), Float64(t * t), 1.0) * fma(x, 0.5, Float64(-y)))) * sqrt(2.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[Sqrt[z], $MachinePrecision] * N[(N[(N[(0.125 * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * 0.5 + (-y)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   3. lift--.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \color{blue}{\sqrt{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   8. sqrt-prod N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)} \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   10. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   11. *-commutative N/A

       \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   12. lift-exp.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{\frac{t \cdot t}{2}}} \]

   13. lift-*.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{t \cdot t}}{2}} \]

   14. lift-/.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{t \cdot t}{2}}} \]

   15. pow2 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{{t}^{2}}}{2}} \]

   16. exp-sqrt N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

   17. pow1/2 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{{\left(e^{{t}^{2}}\right)}^{\frac{1}{2}}} \]

   18. exp-prod N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{{t}^{2} \cdot \frac{1}{2}}} \]

   19. *-commutative N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{1}{2} \cdot {t}^{2}}} \]

3. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\left(\sqrt{z} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2}} \]

4. Taylor expanded in t around 0

   \[\leadsto \left(\left(\sqrt{z} \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \left(\frac{1}{2} \cdot {t}^{2} + \color{blue}{1}\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   2. *-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \left({t}^{2} \cdot \frac{1}{2} + 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left({t}^{2}, \color{blue}{\frac{1}{2}}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   4. pow2 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   5. lift-*.f64 86.1

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]

6. Applied rewrites 86.1%

   \[\leadsto \left(\left(\sqrt{z} \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, 0.5, 1\right)}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right)} \cdot \sqrt{2} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right)} \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\sqrt{z}} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   4. lift-fma.f64 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x + \left(-y\right)\right)}\right) \cdot \sqrt{2} \]

   5. lift-neg.f64 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \cdot \left(\frac{1}{2} \cdot x + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \cdot \sqrt{2} \]

   6. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)} \cdot \sqrt{2} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)} \cdot \sqrt{2} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \left(\color{blue}{\sqrt{z}} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right) \cdot \sqrt{2} \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\sqrt{z} \cdot \color{blue}{\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}\right) \cdot \sqrt{2} \]

   10. *-commutative N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(\color{blue}{x \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right) \cdot \sqrt{2} \]

   11. lift-neg.f64 N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(x \cdot \frac{1}{2} + \color{blue}{\left(-y\right)}\right)\right)\right) \cdot \sqrt{2} \]

   12. lower-fma.f64 87.6

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(x, 0.5, -y\right)}\right)\right) \cdot \sqrt{2} \]

8. Applied rewrites 87.6%

   \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right) \cdot \mathsf{fma}\left(x, 0.5, -y\right)\right)\right)} \cdot \sqrt{2} \]

9. Taylor expanded in t around 0

   \[\leadsto \left(\sqrt{z} \cdot \left(\color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right)\right)} \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\left({t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right) + \color{blue}{1}\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

    2. *-commutative N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\left(\left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right) \cdot {t}^{2} + 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

    3. lower-fma.f64 N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}, \color{blue}{{t}^{2}}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

    4. +-commutative N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(\frac{1}{8} \cdot {t}^{2} + \frac{1}{2}, {\color{blue}{t}}^{2}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

    5. lower-fma.f64 N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, {t}^{2}, \frac{1}{2}\right), {\color{blue}{t}}^{2}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

    6. pow2 N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

    7. lift-*.f64 N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

    8. pow2 N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, t \cdot t, \frac{1}{2}\right), t \cdot \color{blue}{t}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

    9. lift-*.f64 93.5

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.125, t \cdot t, 0.5\right), t \cdot \color{blue}{t}, 1\right) \cdot \mathsf{fma}\left(x, 0.5, -y\right)\right)\right) \cdot \sqrt{2} \]

11. Applied rewrites 93.5%

    \[\leadsto \left(\sqrt{z} \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.125, t \cdot t, 0.5\right), t \cdot t, 1\right)} \cdot \mathsf{fma}\left(x, 0.5, -y\right)\right)\right) \cdot \sqrt{2} \]
12. Add Preprocessing

Alternative 11: 72.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\\ \mathbf{if}\;t \leq 7.8 \cdot 10^{+43}:\\ \;\;\;\;\left(\sqrt{z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t\_1, t \cdot t, 1\right)} \cdot 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (* t t) 0.5 1.0)))
   (if (<= t 7.8e+43)
     (* (* (sqrt z) (fma 0.5 x (- y))) (sqrt 2.0))
     (if (<= t 2e+134)
       (* (* (sqrt (* (* 2.0 z) (fma t_1 (* t t) 1.0))) 0.5) x)
       (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((t * t), 0.5, 1.0);
	double tmp;
	if (t <= 7.8e+43) {
		tmp = (sqrt(z) * fma(0.5, x, -y)) * sqrt(2.0);
	} else if (t <= 2e+134) {
		tmp = (sqrt(((2.0 * z) * fma(t_1, (t * t), 1.0))) * 0.5) * x;
	} else {
		tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(t * t), 0.5, 1.0)
	tmp = 0.0
	if (t <= 7.8e+43)
		tmp = Float64(Float64(sqrt(z) * fma(0.5, x, Float64(-y))) * sqrt(2.0));
	elseif (t <= 2e+134)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * z) * fma(t_1, Float64(t * t), 1.0))) * 0.5) * x);
	else
		tmp = Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, If[LessEqual[t, 7.8e+43], N[(N[(N[Sqrt[z], $MachinePrecision] * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2e+134], N[(N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t$95$1 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.8000000000000001e43

   1. Initial program 99.4%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \color{blue}{\sqrt{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      8. sqrt-prod N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)} \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      12. lift-exp.f64 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{\frac{t \cdot t}{2}}} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{t \cdot t}}{2}} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{t \cdot t}{2}}} \]

      15. pow2 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{{t}^{2}}}{2}} \]

      16. exp-sqrt N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

      17. pow1/2 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{{\left(e^{{t}^{2}}\right)}^{\frac{1}{2}}} \]

      18. exp-prod N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{{t}^{2} \cdot \frac{1}{2}}} \]

      19. *-commutative N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{1}{2} \cdot {t}^{2}}} \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2}} \]

   4. Taylor expanded in t around 0

      \[\leadsto \left(\color{blue}{\sqrt{z}} \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 68.9

         \[\leadsto \left(\sqrt{z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]

   6. Applied rewrites 68.9%

      \[\leadsto \left(\color{blue}{\sqrt{z}} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]

   if 7.8000000000000001e43 < t < 1.99999999999999984e134

   1. Initial program 99.3%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \color{blue}{\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(\left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right) \cdot \color{blue}{x}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{x}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\left(\sqrt{\left(2 \cdot z\right) \cdot {\left(e^{t}\right)}^{t}} \cdot 0.5\right) \cdot x} \]

   5. Taylor expanded in t around 0

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(1 + {t}^{2} \cdot \left(1 + \frac{1}{2} \cdot {t}^{2}\right)\right)} \cdot \frac{1}{2}\right) \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left({t}^{2} \cdot \left(1 + \frac{1}{2} \cdot {t}^{2}\right) + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(\left(1 + \frac{1}{2} \cdot {t}^{2}\right) \cdot {t}^{2} + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(1 + \frac{1}{2} \cdot {t}^{2}, {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      4. +-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\frac{1}{2} \cdot {t}^{2} + 1, {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left({t}^{2} \cdot \frac{1}{2} + 1, {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({t}^{2}, \frac{1}{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      7. pow2 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      9. pow2 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right), t \cdot t, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      10. lift-*.f64 60.2

          \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right), t \cdot t, 1\right)} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 60.2%

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right), t \cdot t, 1\right)} \cdot 0.5\right) \cdot x \]

   if 1.99999999999999984e134 < t

   1. Initial program 99.0%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in t around 0

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\frac{1}{2} \cdot {t}^{2} + \color{blue}{1}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({t}^{2} \cdot \frac{1}{2} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left({t}^{2}, \color{blue}{\frac{1}{2}}, 1\right) \]

      4. pow2 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \]

      5. lift-*.f64 95.7

         \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right) \]

   4. Applied rewrites 95.7%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, 0.5, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 92.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.125, t \cdot t, 0.5\right), t \cdot t, 1\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (* (- (* x 0.5) y) (sqrt (* z 2.0)))
  (fma (fma 0.125 (* t t) 0.5) (* t t) 1.0)))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * fma(fma(0.125, (t * t), 0.5), (t * t), 1.0);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * fma(fma(0.125, Float64(t * t), 0.5), Float64(t * t), 1.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.125 * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

2. Taylor expanded in t around 0

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right)\right)} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right) + \color{blue}{1}\right) \]

   2. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right) \cdot {t}^{2} + 1\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}, \color{blue}{{t}^{2}}, 1\right) \]

   4. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\frac{1}{8} \cdot {t}^{2} + \frac{1}{2}, {\color{blue}{t}}^{2}, 1\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, {t}^{2}, \frac{1}{2}\right), {\color{blue}{t}}^{2}, 1\right) \]

   6. pow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   8. pow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, t \cdot t, \frac{1}{2}\right), t \cdot \color{blue}{t}, 1\right) \]

   9. lift-*.f64 92.3

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.125, t \cdot t, 0.5\right), t \cdot \color{blue}{t}, 1\right) \]

4. Applied rewrites 92.3%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.125, t \cdot t, 0.5\right), t \cdot t, 1\right)} \]
5. Add Preprocessing

Alternative 13: 87.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right) \cdot \mathsf{fma}\left(x, 0.5, -y\right)\right)\right) \cdot \sqrt{2} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (sqrt z) (* (fma (* t t) 0.5 1.0) (fma x 0.5 (- y)))) (sqrt 2.0)))

C

double code(double x, double y, double z, double t) {
	return (sqrt(z) * (fma((t * t), 0.5, 1.0) * fma(x, 0.5, -y))) * sqrt(2.0);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(sqrt(z) * Float64(fma(Float64(t * t), 0.5, 1.0) * fma(x, 0.5, Float64(-y)))) * sqrt(2.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[Sqrt[z], $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(x * 0.5 + (-y)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   3. lift--.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \color{blue}{\sqrt{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   8. sqrt-prod N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)} \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   10. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   11. *-commutative N/A

       \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   12. lift-exp.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{\frac{t \cdot t}{2}}} \]

   13. lift-*.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{t \cdot t}}{2}} \]

   14. lift-/.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{t \cdot t}{2}}} \]

   15. pow2 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{{t}^{2}}}{2}} \]

   16. exp-sqrt N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

   17. pow1/2 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{{\left(e^{{t}^{2}}\right)}^{\frac{1}{2}}} \]

   18. exp-prod N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{{t}^{2} \cdot \frac{1}{2}}} \]

   19. *-commutative N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{1}{2} \cdot {t}^{2}}} \]

3. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\left(\sqrt{z} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2}} \]

4. Taylor expanded in t around 0

   \[\leadsto \left(\left(\sqrt{z} \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \left(\frac{1}{2} \cdot {t}^{2} + \color{blue}{1}\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   2. *-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \left({t}^{2} \cdot \frac{1}{2} + 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left({t}^{2}, \color{blue}{\frac{1}{2}}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   4. pow2 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   5. lift-*.f64 86.1

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]

6. Applied rewrites 86.1%

   \[\leadsto \left(\left(\sqrt{z} \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, 0.5, 1\right)}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right)} \cdot \sqrt{2} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right)} \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\sqrt{z}} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   4. lift-fma.f64 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x + \left(-y\right)\right)}\right) \cdot \sqrt{2} \]

   5. lift-neg.f64 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \cdot \left(\frac{1}{2} \cdot x + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \cdot \sqrt{2} \]

   6. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)} \cdot \sqrt{2} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)} \cdot \sqrt{2} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \left(\color{blue}{\sqrt{z}} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right) \cdot \sqrt{2} \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\sqrt{z} \cdot \color{blue}{\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}\right) \cdot \sqrt{2} \]

   10. *-commutative N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(\color{blue}{x \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right) \cdot \sqrt{2} \]

   11. lift-neg.f64 N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(x \cdot \frac{1}{2} + \color{blue}{\left(-y\right)}\right)\right)\right) \cdot \sqrt{2} \]

   12. lower-fma.f64 87.6

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(x, 0.5, -y\right)}\right)\right) \cdot \sqrt{2} \]

8. Applied rewrites 87.6%

   \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right) \cdot \mathsf{fma}\left(x, 0.5, -y\right)\right)\right)} \cdot \sqrt{2} \]
9. Add Preprocessing

Alternative 14: 87.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{z} \cdot \left(\left(\mathsf{fma}\left(0.5, x, -y\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\right) \cdot \sqrt{2}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (sqrt z) (* (* (fma 0.5 x (- y)) (fma (* t t) 0.5 1.0)) (sqrt 2.0))))

C

double code(double x, double y, double z, double t) {
	return sqrt(z) * ((fma(0.5, x, -y) * fma((t * t), 0.5, 1.0)) * sqrt(2.0));
}

Julia

function code(x, y, z, t)
	return Float64(sqrt(z) * Float64(Float64(fma(0.5, x, Float64(-y)) * fma(Float64(t * t), 0.5, 1.0)) * sqrt(2.0)))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Sqrt[z], $MachinePrecision] * N[(N[(N[(0.5 * x + (-y)), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   3. lift--.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \color{blue}{\sqrt{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   8. sqrt-prod N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)} \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   10. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   11. *-commutative N/A

       \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   12. lift-exp.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{\frac{t \cdot t}{2}}} \]

   13. lift-*.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{t \cdot t}}{2}} \]

   14. lift-/.f64 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{t \cdot t}{2}}} \]

   15. pow2 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{{t}^{2}}}{2}} \]

   16. exp-sqrt N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

   17. pow1/2 N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{{\left(e^{{t}^{2}}\right)}^{\frac{1}{2}}} \]

   18. exp-prod N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{{t}^{2} \cdot \frac{1}{2}}} \]

   19. *-commutative N/A

       \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{1}{2} \cdot {t}^{2}}} \]

3. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\left(\sqrt{z} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2}} \]

4. Taylor expanded in t around 0

   \[\leadsto \left(\left(\sqrt{z} \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \left(\frac{1}{2} \cdot {t}^{2} + \color{blue}{1}\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   2. *-commutative N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \left({t}^{2} \cdot \frac{1}{2} + 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left({t}^{2}, \color{blue}{\frac{1}{2}}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   4. pow2 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   5. lift-*.f64 86.1

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]

6. Applied rewrites 86.1%

   \[\leadsto \left(\left(\sqrt{z} \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, 0.5, 1\right)}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right)} \cdot \sqrt{2} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right)} \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\sqrt{z}} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]

   4. lift-fma.f64 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x + \left(-y\right)\right)}\right) \cdot \sqrt{2} \]

   5. lift-neg.f64 N/A

      \[\leadsto \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \cdot \left(\frac{1}{2} \cdot x + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \cdot \sqrt{2} \]

   6. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)} \cdot \sqrt{2} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)} \cdot \sqrt{2} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \left(\color{blue}{\sqrt{z}} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right) \cdot \sqrt{2} \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\sqrt{z} \cdot \color{blue}{\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}\right) \cdot \sqrt{2} \]

   10. *-commutative N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(\color{blue}{x \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right) \cdot \sqrt{2} \]

   11. lift-neg.f64 N/A

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \left(x \cdot \frac{1}{2} + \color{blue}{\left(-y\right)}\right)\right)\right) \cdot \sqrt{2} \]

   12. lower-fma.f64 87.6

       \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(x, 0.5, -y\right)}\right)\right) \cdot \sqrt{2} \]

8. Applied rewrites 87.6%

   \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right) \cdot \mathsf{fma}\left(x, 0.5, -y\right)\right)\right)} \cdot \sqrt{2} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2}} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right)} \cdot \sqrt{2} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \left(\color{blue}{\sqrt{z}} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \sqrt{2} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{z} \cdot \left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right)\right) \cdot \color{blue}{\sqrt{2}} \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right) \cdot \sqrt{2}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right) \cdot \sqrt{2}\right)} \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{z}} \cdot \left(\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right) \cdot \sqrt{2}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \sqrt{z} \cdot \color{blue}{\left(\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, -y\right)\right) \cdot \sqrt{2}\right)} \]

10. Applied rewrites 87.6%

    \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\left(\mathsf{fma}\left(0.5, x, -y\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\right) \cdot \sqrt{2}\right)} \]
11. Add Preprocessing

Alternative 15: 71.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt{z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t \cdot t\right) \cdot \left(\sqrt{2 \cdot z} \cdot \left(x \cdot 0.5 - y\right)\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 6.2e-7)
   (* (* (sqrt z) (fma 0.5 x (- y))) (sqrt 2.0))
   (* (* (* t t) (* (sqrt (* 2.0 z)) (- (* x 0.5) y))) 0.5)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 6.2e-7) {
		tmp = (sqrt(z) * fma(0.5, x, -y)) * sqrt(2.0);
	} else {
		tmp = ((t * t) * (sqrt((2.0 * z)) * ((x * 0.5) - y))) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 6.2e-7)
		tmp = Float64(Float64(sqrt(z) * fma(0.5, x, Float64(-y))) * sqrt(2.0));
	else
		tmp = Float64(Float64(Float64(t * t) * Float64(sqrt(Float64(2.0 * z)) * Float64(Float64(x * 0.5) - y))) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 6.2e-7], N[(N[(N[Sqrt[z], $MachinePrecision] * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * t), $MachinePrecision] * N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.1999999999999999e-7

   1. Initial program 99.4%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \color{blue}{\sqrt{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      8. sqrt-prod N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)} \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      12. lift-exp.f64 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{\frac{t \cdot t}{2}}} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{t \cdot t}}{2}} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{t \cdot t}{2}}} \]

      15. pow2 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{{t}^{2}}}{2}} \]

      16. exp-sqrt N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

      17. pow1/2 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{{\left(e^{{t}^{2}}\right)}^{\frac{1}{2}}} \]

      18. exp-prod N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{{t}^{2} \cdot \frac{1}{2}}} \]

      19. *-commutative N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{1}{2} \cdot {t}^{2}}} \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2}} \]

   4. Taylor expanded in t around 0

      \[\leadsto \left(\color{blue}{\sqrt{z}} \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 71.6

         \[\leadsto \left(\sqrt{z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]

   6. Applied rewrites 71.6%

      \[\leadsto \left(\color{blue}{\sqrt{z}} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]

   if 6.1999999999999999e-7 < t

   1. Initial program 99.2%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) \cdot \sqrt{z}\right) + \sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)\right) \cdot \sqrt{z} + \color{blue}{\sqrt{z}} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)\right) \cdot \sqrt{z} + \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \color{blue}{\sqrt{z}} \]

      3. distribute-rgt-out N/A

         \[\leadsto \sqrt{z} \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) + \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{z} \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) + \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)} + \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) + \color{blue}{\sqrt{2}} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{z} \cdot \mathsf{fma}\left(\frac{1}{2} \cdot {t}^{2}, \color{blue}{\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)}, \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{\sqrt{z} \cdot \mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.5, \sqrt{2} \cdot \mathsf{fma}\left(0.5, x, -y\right), \sqrt{2} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) \cdot \sqrt{z}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) \cdot \sqrt{z}\right) \cdot \frac{1}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) \cdot \sqrt{z}\right) \cdot \frac{1}{2} \]

   7. Applied rewrites 72.1%

      \[\leadsto \left(\left(t \cdot t\right) \cdot \left(\sqrt{2 \cdot z} \cdot \left(x \cdot 0.5 - y\right)\right)\right) \cdot \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 66.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{+43}:\\ \;\;\;\;\left(\sqrt{z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 8e+43)
   (* (* (sqrt z) (fma 0.5 x (- y))) (sqrt 2.0))
   (* (* (sqrt (* (* 2.0 z) (fma t t 1.0))) 0.5) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 8e+43) {
		tmp = (sqrt(z) * fma(0.5, x, -y)) * sqrt(2.0);
	} else {
		tmp = (sqrt(((2.0 * z) * fma(t, t, 1.0))) * 0.5) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 8e+43)
		tmp = Float64(Float64(sqrt(z) * fma(0.5, x, Float64(-y))) * sqrt(2.0));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * z) * fma(t, t, 1.0))) * 0.5) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 8e+43], N[(N[(N[Sqrt[z], $MachinePrecision] * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.00000000000000011e43

   1. Initial program 99.4%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \color{blue}{\sqrt{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      8. sqrt-prod N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)} \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      12. lift-exp.f64 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{\frac{t \cdot t}{2}}} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{t \cdot t}}{2}} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{t \cdot t}{2}}} \]

      15. pow2 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\frac{\color{blue}{{t}^{2}}}{2}} \]

      16. exp-sqrt N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

      17. pow1/2 N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{{\left(e^{{t}^{2}}\right)}^{\frac{1}{2}}} \]

      18. exp-prod N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot \color{blue}{e^{{t}^{2} \cdot \frac{1}{2}}} \]

      19. *-commutative N/A

          \[\leadsto \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z}\right) \cdot e^{\color{blue}{\frac{1}{2} \cdot {t}^{2}}} \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2}} \]

   4. Taylor expanded in t around 0

      \[\leadsto \left(\color{blue}{\sqrt{z}} \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \cdot \sqrt{2} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 68.9

         \[\leadsto \left(\sqrt{z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]

   6. Applied rewrites 68.9%

      \[\leadsto \left(\color{blue}{\sqrt{z}} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \cdot \sqrt{2} \]

   if 8.00000000000000011e43 < t

   1. Initial program 99.1%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \color{blue}{\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(\left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right) \cdot \color{blue}{x}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{x}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\left(\sqrt{\left(2 \cdot z\right) \cdot {\left(e^{t}\right)}^{t}} \cdot 0.5\right) \cdot x} \]

   5. Taylor expanded in t around 0

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(1 + {t}^{2}\right)} \cdot \frac{1}{2}\right) \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left({t}^{2} + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      2. pow2 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(t \cdot t + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      3. lower-fma.f64 57.0

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 57.0%

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot 0.5\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 66.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{+43}:\\ \;\;\;\;\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 8e+43)
   (* (* (sqrt z) (sqrt 2.0)) (fma 0.5 x (- y)))
   (* (* (sqrt (* (* 2.0 z) (fma t t 1.0))) 0.5) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 8e+43) {
		tmp = (sqrt(z) * sqrt(2.0)) * fma(0.5, x, -y);
	} else {
		tmp = (sqrt(((2.0 * z) * fma(t, t, 1.0))) * 0.5) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 8e+43)
		tmp = Float64(Float64(sqrt(z) * sqrt(2.0)) * fma(0.5, x, Float64(-y)));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * z) * fma(t, t, 1.0))) * 0.5) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 8e+43], N[(N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.00000000000000011e43

   1. Initial program 99.4%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x - y\right)} \]

      2. sqrt-prod N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - y\right) \]

      11. *-lft-identity N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - 1 \cdot \color{blue}{y}\right) \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right) \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x + \color{blue}{-1 \cdot y}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{x}, -1 \cdot y\right) \]

      15. mul-1-neg N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \]

      16. lower-neg.f64 69.1

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right) \]

   4. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{2}}, x, -y\right) \]

      3. sqrt-prod N/A

         \[\leadsto \left(\sqrt{2} \cdot \sqrt{z}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{2}}, x, -y\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{2}}, x, -y\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{2}}, x, -y\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right) \]

      7. lift-sqrt.f64 68.9

         \[\leadsto \left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(0.5, x, -y\right) \]

   6. Applied rewrites 68.9%

      \[\leadsto \left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\color{blue}{0.5}, x, -y\right) \]

   if 8.00000000000000011e43 < t

   1. Initial program 99.1%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \color{blue}{\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(\left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right) \cdot \color{blue}{x}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{x}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\left(\sqrt{\left(2 \cdot z\right) \cdot {\left(e^{t}\right)}^{t}} \cdot 0.5\right) \cdot x} \]

   5. Taylor expanded in t around 0

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(1 + {t}^{2}\right)} \cdot \frac{1}{2}\right) \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left({t}^{2} + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      2. pow2 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(t \cdot t + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      3. lower-fma.f64 57.0

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 57.0%

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot 0.5\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 66.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{+43}:\\ \;\;\;\;\sqrt{z} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 8e+43)
   (* (sqrt z) (* (- (* x 0.5) y) (sqrt 2.0)))
   (* (* (sqrt (* (* 2.0 z) (fma t t 1.0))) 0.5) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 8e+43) {
		tmp = sqrt(z) * (((x * 0.5) - y) * sqrt(2.0));
	} else {
		tmp = (sqrt(((2.0 * z) * fma(t, t, 1.0))) * 0.5) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 8e+43)
		tmp = Float64(sqrt(z) * Float64(Float64(Float64(x * 0.5) - y) * sqrt(2.0)));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * z) * fma(t, t, 1.0))) * 0.5) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 8e+43], N[(N[Sqrt[z], $MachinePrecision] * N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.00000000000000011e43

   1. Initial program 99.4%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) \cdot \sqrt{z}\right) + \sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)\right) \cdot \sqrt{z} + \color{blue}{\sqrt{z}} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)\right) \cdot \sqrt{z} + \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \color{blue}{\sqrt{z}} \]

      3. distribute-rgt-out N/A

         \[\leadsto \sqrt{z} \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) + \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{z} \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) + \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)} + \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) + \color{blue}{\sqrt{2}} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{z} \cdot \mathsf{fma}\left(\frac{1}{2} \cdot {t}^{2}, \color{blue}{\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)}, \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

   4. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\sqrt{z} \cdot \mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.5, \sqrt{2} \cdot \mathsf{fma}\left(0.5, x, -y\right), \sqrt{2} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right)} \]

   5. Taylor expanded in t around 0

      \[\leadsto \sqrt{z} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot x - y\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\frac{1}{2} \cdot x - y\right) \cdot \sqrt{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\frac{1}{2} \cdot x - y\right) \cdot \sqrt{2}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\frac{1}{2} \cdot x - y\right) \cdot \sqrt{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{2}\right) \]

      6. lift-sqrt.f64 68.9

         \[\leadsto \sqrt{z} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right) \]

   7. Applied rewrites 68.9%

      \[\leadsto \sqrt{z} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{2}}\right) \]

   if 8.00000000000000011e43 < t

   1. Initial program 99.1%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \color{blue}{\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(\left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right) \cdot \color{blue}{x}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{x}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\left(\sqrt{\left(2 \cdot z\right) \cdot {\left(e^{t}\right)}^{t}} \cdot 0.5\right) \cdot x} \]

   5. Taylor expanded in t around 0

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(1 + {t}^{2}\right)} \cdot \frac{1}{2}\right) \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left({t}^{2} + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      2. pow2 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(t \cdot t + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      3. lower-fma.f64 57.0

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 57.0%

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot 0.5\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 85.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (fma (* t t) 0.5 1.0)))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * fma((t * t), 0.5, 1.0);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * fma(Float64(t * t), 0.5, 1.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

2. Taylor expanded in t around 0

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\frac{1}{2} \cdot {t}^{2} + \color{blue}{1}\right) \]

   2. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({t}^{2} \cdot \frac{1}{2} + 1\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left({t}^{2}, \color{blue}{\frac{1}{2}}, 1\right) \]

   4. pow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \]

   5. lift-*.f64 85.5

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right) \]

4. Applied rewrites 85.5%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, 0.5, 1\right)} \]
5. Add Preprocessing

Alternative 20: 66.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{+43}:\\ \;\;\;\;\sqrt{z + z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 8e+43)
   (* (sqrt (+ z z)) (fma 0.5 x (- y)))
   (* (* (sqrt (* (* 2.0 z) (fma t t 1.0))) 0.5) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 8e+43) {
		tmp = sqrt((z + z)) * fma(0.5, x, -y);
	} else {
		tmp = (sqrt(((2.0 * z) * fma(t, t, 1.0))) * 0.5) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 8e+43)
		tmp = Float64(sqrt(Float64(z + z)) * fma(0.5, x, Float64(-y)));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * z) * fma(t, t, 1.0))) * 0.5) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 8e+43], N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.00000000000000011e43

   1. Initial program 99.4%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x - y\right)} \]

      2. sqrt-prod N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - y\right) \]

      11. *-lft-identity N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - 1 \cdot \color{blue}{y}\right) \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right) \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x + \color{blue}{-1 \cdot y}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{x}, -1 \cdot y\right) \]

      15. mul-1-neg N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \]

      16. lower-neg.f64 69.1

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right) \]

   4. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right) \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{z + z} \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right) \]

      3. lower-+.f64 69.1

         \[\leadsto \sqrt{z + z} \cdot \mathsf{fma}\left(0.5, x, -y\right) \]

   6. Applied rewrites 69.1%

      \[\leadsto \sqrt{z + z} \cdot \mathsf{fma}\left(0.5, x, -y\right) \]

   if 8.00000000000000011e43 < t

   1. Initial program 99.1%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \color{blue}{\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(\left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right) \cdot \color{blue}{x}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{x}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\left(\sqrt{\left(2 \cdot z\right) \cdot {\left(e^{t}\right)}^{t}} \cdot 0.5\right) \cdot x} \]

   5. Taylor expanded in t around 0

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(1 + {t}^{2}\right)} \cdot \frac{1}{2}\right) \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left({t}^{2} + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      2. pow2 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(t \cdot t + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      3. lower-fma.f64 57.0

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 57.0%

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot 0.5\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 43.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z + z} \cdot \left(-y\right)\\ \mathbf{if}\;y \leq -9.8 \cdot 10^{-48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{-94}:\\ \;\;\;\;\left(\sqrt{2 \cdot z} \cdot 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (sqrt (+ z z)) (- y))))
   (if (<= y -9.8e-48)
     t_1
     (if (<= y 1.18e-94) (* (* (sqrt (* 2.0 z)) 0.5) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + z)) * -y;
	double tmp;
	if (y <= -9.8e-48) {
		tmp = t_1;
	} else if (y <= 1.18e-94) {
		tmp = (sqrt((2.0 * z)) * 0.5) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z + z)) * -y
    if (y <= (-9.8d-48)) then
        tmp = t_1
    else if (y <= 1.18d-94) then
        tmp = (sqrt((2.0d0 * z)) * 0.5d0) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + z)) * -y;
	double tmp;
	if (y <= -9.8e-48) {
		tmp = t_1;
	} else if (y <= 1.18e-94) {
		tmp = (Math.sqrt((2.0 * z)) * 0.5) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z + z)) * -y
	tmp = 0
	if y <= -9.8e-48:
		tmp = t_1
	elif y <= 1.18e-94:
		tmp = (math.sqrt((2.0 * z)) * 0.5) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + z)) * Float64(-y))
	tmp = 0.0
	if (y <= -9.8e-48)
		tmp = t_1;
	elseif (y <= 1.18e-94)
		tmp = Float64(Float64(sqrt(Float64(2.0 * z)) * 0.5) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + z)) * -y;
	tmp = 0.0;
	if (y <= -9.8e-48)
		tmp = t_1;
	elseif (y <= 1.18e-94)
		tmp = (sqrt((2.0 * z)) * 0.5) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]}, If[LessEqual[y, -9.8e-48], t$95$1, If[LessEqual[y, 1.18e-94], N[(N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.8000000000000005e-48 or 1.18e-94 < y

   1. Initial program 99.8%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x - y\right)} \]

      2. sqrt-prod N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - y\right) \]

      11. *-lft-identity N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - 1 \cdot \color{blue}{y}\right) \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right) \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x + \color{blue}{-1 \cdot y}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{x}, -1 \cdot y\right) \]

      15. mul-1-neg N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \]

      16. lower-neg.f64 59.6

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right) \]

   4. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-1 \cdot \color{blue}{y}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\mathsf{neg}\left(y\right)\right) \]

      2. lift-neg.f64 42.3

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(-y\right) \]

   7. Applied rewrites 42.3%

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-y\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(-y\right) \]

      2. count-2-rev N/A

         \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]

      3. lower-+.f64 42.3

         \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]

   9. Applied rewrites 42.3%

      \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]

   if -9.8000000000000005e-48 < y < 1.18e-94

   1. Initial program 98.6%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \color{blue}{\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(\left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right) \cdot \color{blue}{x}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{x}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

   4. Applied rewrites 85.0%

      \[\leadsto \color{blue}{\left(\sqrt{\left(2 \cdot z\right) \cdot {\left(e^{t}\right)}^{t}} \cdot 0.5\right) \cdot x} \]

   5. Taylor expanded in t around 0

      \[\leadsto \left(\sqrt{2 \cdot z} \cdot \frac{1}{2}\right) \cdot x \]
   6. Step-by-step derivation

      1. lift-*.f64 44.8

         \[\leadsto \left(\sqrt{2 \cdot z} \cdot 0.5\right) \cdot x \]

   7. Applied rewrites 44.8%

      \[\leadsto \left(\sqrt{2 \cdot z} \cdot 0.5\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 57.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{z + z} \cdot \mathsf{fma}\left(0.5, x, -y\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (sqrt (+ z z)) (fma 0.5 x (- y))))

C

double code(double x, double y, double z, double t) {
	return sqrt((z + z)) * fma(0.5, x, -y);
}

Julia

function code(x, y, z, t)
	return Float64(sqrt(Float64(z + z)) * fma(0.5, x, Float64(-y)))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

2. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
3. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x - y\right)} \]

   2. sqrt-prod N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right) \]

   3. *-commutative N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

   8. *-commutative N/A

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

   10. *-commutative N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - y\right) \]

   11. *-lft-identity N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - 1 \cdot \color{blue}{y}\right) \]

   12. metadata-eval N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right) \]

   13. fp-cancel-sign-sub-inv N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x + \color{blue}{-1 \cdot y}\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{x}, -1 \cdot y\right) \]

   15. mul-1-neg N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \]

   16. lower-neg.f64 57.5

       \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right) \]

4. Applied rewrites 57.5%

   \[\leadsto \color{blue}{\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right) \]

   2. count-2-rev N/A

      \[\leadsto \sqrt{z + z} \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right) \]

   3. lower-+.f64 57.5

      \[\leadsto \sqrt{z + z} \cdot \mathsf{fma}\left(0.5, x, -y\right) \]

6. Applied rewrites 57.5%

   \[\leadsto \sqrt{z + z} \cdot \mathsf{fma}\left(0.5, x, -y\right) \]
7. Add Preprocessing

Alternative 23: 30.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{z + z} \cdot \left(-y\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (sqrt (+ z z)) (- y)))

C

double code(double x, double y, double z, double t) {
	return sqrt((z + z)) * -y;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt((z + z)) * -y
end function

Java

public static double code(double x, double y, double z, double t) {
	return Math.sqrt((z + z)) * -y;
}

Python

def code(x, y, z, t):
	return math.sqrt((z + z)) * -y

Julia

function code(x, y, z, t)
	return Float64(sqrt(Float64(z + z)) * Float64(-y))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = sqrt((z + z)) * -y;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

2. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
3. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x - y\right)} \]

   2. sqrt-prod N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right) \]

   3. *-commutative N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

   8. *-commutative N/A

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

   10. *-commutative N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - y\right) \]

   11. *-lft-identity N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - 1 \cdot \color{blue}{y}\right) \]

   12. metadata-eval N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right) \]

   13. fp-cancel-sign-sub-inv N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x + \color{blue}{-1 \cdot y}\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{x}, -1 \cdot y\right) \]

   15. mul-1-neg N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \]

   16. lower-neg.f64 57.5

       \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right) \]

4. Applied rewrites 57.5%

   \[\leadsto \color{blue}{\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto \sqrt{2 \cdot z} \cdot \left(-1 \cdot \color{blue}{y}\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(\mathsf{neg}\left(y\right)\right) \]

   2. lift-neg.f64 30.7

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-y\right) \]

7. Applied rewrites 30.7%

   \[\leadsto \sqrt{2 \cdot z} \cdot \left(-y\right) \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-y\right) \]

   2. count-2-rev N/A

      \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]

   3. lower-+.f64 30.7

      \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]

9. Applied rewrites 30.7%

   \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]
10. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(e^{1}\right)}^{\left(\frac{t \cdot t}{2}\right)} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (pow (exp 1.0) (/ (* t t) 2.0))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * pow(exp(1.0), ((t * t) / 2.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * (exp(1.0d0) ** ((t * t) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.pow(Math.exp(1.0), ((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.pow(math.exp(1.0), ((t * t) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * (exp(1.0) ^ Float64(Float64(t * t) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * (exp(1.0) ^ ((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[1.0], $MachinePrecision], N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025095 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :alt
  (! :herbie-platform default (* (* (- (* x 1/2) y) (sqrt (* z 2))) (pow (exp 1) (/ (* t t) 2))))

  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))