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

Percentage Accurate: 99.5% → 99.5%

Time: 7.3s

Alternatives: 12

Speedup: 1.0×

• 44.9% 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: 99.5% 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: 99.5% 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 99.9%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Add Preprocessing
3. 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 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 58 \lor \neg \left(t \leq 1.9 \cdot 10^{+47}\right):\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z + z}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot {\left(1 + t\right)}^{t}} \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t 58.0) (not (<= t 1.9e+47)))
   (*
    (* (- (* x 0.5) y) (sqrt (+ z z)))
    (fma
     (fma (fma 0.020833333333333332 (* t t) 0.125) (* t t) 0.5)
     (* t t)
     1.0))
   (* (sqrt (* (* 2.0 z) (pow (+ 1.0 t) t))) (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= 58.0) || !(t <= 1.9e+47)) {
		tmp = (((x * 0.5) - y) * sqrt((z + z))) * fma(fma(fma(0.020833333333333332, (t * t), 0.125), (t * t), 0.5), (t * t), 1.0);
	} else {
		tmp = sqrt(((2.0 * z) * pow((1.0 + t), t))) * -y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= 58.0) || !(t <= 1.9e+47))
		tmp = Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z + z))) * fma(fma(fma(0.020833333333333332, Float64(t * t), 0.125), Float64(t * t), 0.5), Float64(t * t), 1.0));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * z) * (Float64(1.0 + t) ^ t))) * Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, 58.0], N[Not[LessEqual[t, 1.9e+47]], $MachinePrecision]], N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z + z), $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], N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Power[N[(1.0 + t), $MachinePrecision], t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 58 or 1.9000000000000002e47 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)} \]
   4. 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 96.7

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

   5. Applied rewrites 96.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 96.7

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

   7. Applied rewrites 96.7%

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

   if 58 < t < 1.9000000000000002e47

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 81.8%

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

   6. Taylor expanded in t around 0

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

      1. lower-+.f64 81.8

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

   8. Applied rewrites 81.8%

      \[\leadsto \left(-\sqrt{\left(2 \cdot z\right) \cdot {\left(1 + t\right)}^{t}}\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 58 \lor \neg \left(t \leq 1.9 \cdot 10^{+47}\right):\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z + z}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot {\left(1 + t\right)}^{t}} \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.0% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (* (- (* x 0.5) y) (sqrt (+ z z)))
  (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 + z))) * 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 + z))) * 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 + z), $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 99.9%

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

3. Taylor expanded in t around 0

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)} \]
4. 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 94.0

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

5. Applied rewrites 94.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f64 94.0

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

7. Applied rewrites 94.0%

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

Alternative 4: 86.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 110000000000:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot 0.5\right) \cdot \left(\sqrt{2} \cdot \left(0.5 \cdot x - y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 110000000000.0)
   (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (fma (* t t) 0.5 1.0))
   (* (sqrt z) (* (* (* t t) 0.5) (* (sqrt 2.0) (- (* 0.5 x) y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 110000000000.0) {
		tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * fma((t * t), 0.5, 1.0);
	} else {
		tmp = sqrt(z) * (((t * t) * 0.5) * (sqrt(2.0) * ((0.5 * x) - y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 110000000000.0)
		tmp = Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * fma(Float64(t * t), 0.5, 1.0));
	else
		tmp = Float64(sqrt(z) * Float64(Float64(Float64(t * t) * 0.5) * Float64(sqrt(2.0) * Float64(Float64(0.5 * x) - y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 110000000000.0], 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], N[(N[Sqrt[z], $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.1e11

   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. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)} \]
   4. 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 89.7

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

   5. Applied rewrites 89.7%

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

   if 1.1e11 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) \cdot \sqrt{z}\right) + \sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   4. 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

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{z} \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)}\right) \]
   7. 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

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

      8. lower-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 74.3

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

   8. Applied rewrites 74.3%

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

Alternative 5: 71.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z + z}\\ \mathbf{if}\;t \leq 2300:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(0.5, x, -y\right)\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+78}:\\ \;\;\;\;t\_1 \cdot \left(\left(\frac{x}{y} \cdot 0.5 - 1\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t \cdot t\right) \cdot \left(\sqrt{2 \cdot z} \cdot \left(0.5 \cdot x - y\right)\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z z))))
   (if (<= t 2300.0)
     (* t_1 (fma 0.5 x (- y)))
     (if (<= t 4e+78)
       (* t_1 (* (- (* (/ x y) 0.5) 1.0) y))
       (* (* (* t t) (* (sqrt (* 2.0 z)) (- (* 0.5 x) y))) 0.5)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + z));
	double tmp;
	if (t <= 2300.0) {
		tmp = t_1 * fma(0.5, x, -y);
	} else if (t <= 4e+78) {
		tmp = t_1 * ((((x / y) * 0.5) - 1.0) * y);
	} else {
		tmp = ((t * t) * (sqrt((2.0 * z)) * ((0.5 * x) - y))) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z + z))
	tmp = 0.0
	if (t <= 2300.0)
		tmp = Float64(t_1 * fma(0.5, x, Float64(-y)));
	elseif (t <= 4e+78)
		tmp = Float64(t_1 * Float64(Float64(Float64(Float64(x / y) * 0.5) - 1.0) * y));
	else
		tmp = Float64(Float64(Float64(t * t) * Float64(sqrt(Float64(2.0 * z)) * Float64(Float64(0.5 * x) - y))) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 2300.0], N[(t$95$1 * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4e+78], N[(t$95$1 * N[(N[(N[(N[(x / y), $MachinePrecision] * 0.5), $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * t), $MachinePrecision] * N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 2300

   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. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   4. 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 69.8

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-1 \cdot \color{blue}{y}\right) \]
   7. 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 40.3

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

   8. Applied rewrites 40.3%

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-y\right) \]
   9. 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 40.3

         \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]

   10. Applied rewrites 40.3%

       \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]

   11. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. lift-neg.f64 N/A

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

       3. +-commutative N/A

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

       4. lift-fma.f64 69.8

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

   13. Applied rewrites 69.8%

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

   if 2300 < t < 4.00000000000000003e78

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   4. 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 10.3

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

   5. Applied rewrites 10.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-1 \cdot \color{blue}{y}\right) \]
   7. 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 3.3

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

   8. Applied rewrites 3.3%

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-y\right) \]
   9. 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 3.3

         \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]

   10. Applied rewrites 3.3%

       \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]

   11. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 27.0

          \[\leadsto \sqrt{z + z} \cdot \left(\left(\frac{x}{y} \cdot 0.5 - 1\right) \cdot y\right) \]

   13. Applied rewrites 27.0%

       \[\leadsto \sqrt{z + z} \cdot \left(\left(\frac{x}{y} \cdot 0.5 - 1\right) \cdot \color{blue}{y}\right) \]

   if 4.00000000000000003e78 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) \cdot \sqrt{z}\right) + \sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   4. 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

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in t around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) \cdot \sqrt{z}\right)} \]
   7. 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

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

   8. Applied rewrites 91.3%

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

Alternative 6: 92.4% 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 99.9%

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

3. Taylor expanded in t around 0

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right)\right)} \]
4. 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 92.1

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

5. Applied rewrites 92.1%

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

Alternative 7: 69.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{z + z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t, t, 2\right), t \cdot t, 2\right) \cdot z} \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.4e+24)
   (* (sqrt (+ z z)) (fma 0.5 x (- y)))
   (* (sqrt (* (fma (fma t t 2.0) (* t t) 2.0) z)) (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.4e+24) {
		tmp = sqrt((z + z)) * fma(0.5, x, -y);
	} else {
		tmp = sqrt((fma(fma(t, t, 2.0), (t * t), 2.0) * z)) * -y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.4e+24)
		tmp = Float64(sqrt(Float64(z + z)) * fma(0.5, x, Float64(-y)));
	else
		tmp = Float64(sqrt(Float64(fma(fma(t, t, 2.0), Float64(t * t), 2.0) * z)) * Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.4e+24], N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(t * t + 2.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 2.0), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.4000000000000001e24

   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. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   4. 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 67.9

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

   5. Applied rewrites 67.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-1 \cdot \color{blue}{y}\right) \]
   7. 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 38.9

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

   8. Applied rewrites 38.9%

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-y\right) \]
   9. 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 38.9

         \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]

   10. Applied rewrites 38.9%

       \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]

   11. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. lift-neg.f64 N/A

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

       3. +-commutative N/A

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

       4. lift-fma.f64 67.9

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

   13. Applied rewrites 67.9%

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

   if 1.4000000000000001e24 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in t around 0

      \[\leadsto \left(-\sqrt{2 \cdot z + {t}^{2} \cdot \left(2 \cdot z + {t}^{2} \cdot z\right)}\right) \cdot y \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 64.8

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

   8. Applied rewrites 64.8%

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

   10. Applied rewrites 67.2%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{z + z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t, t, 2\right), t \cdot t, 2\right) \cdot z} \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 99.9%

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

3. Taylor expanded in t around 0

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)} \]
4. 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 85.7

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

5. Applied rewrites 85.7%

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

Alternative 9: 66.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{z + z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 9e+67)
   (* (sqrt (+ z z)) (fma 0.5 x (- y)))
   (* (sqrt (* (* 2.0 z) (fma t t 1.0))) (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 9e+67) {
		tmp = sqrt((z + z)) * fma(0.5, x, -y);
	} else {
		tmp = sqrt(((2.0 * z) * fma(t, t, 1.0))) * -y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 9e+67)
		tmp = Float64(sqrt(Float64(z + z)) * fma(0.5, x, Float64(-y)));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * z) * fma(t, t, 1.0))) * Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 9e+67], N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.9999999999999997e67

   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. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   4. 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 65.8

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

   5. Applied rewrites 65.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-1 \cdot \color{blue}{y}\right) \]
   7. 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 37.8

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

   8. Applied rewrites 37.8%

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-y\right) \]
   9. 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 37.8

         \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]

   10. Applied rewrites 37.8%

       \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]

   11. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. lift-neg.f64 N/A

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

       3. +-commutative N/A

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

       4. lift-fma.f64 65.8

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

   13. Applied rewrites 65.8%

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

   if 8.9999999999999997e67 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 77.1%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 66.1

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

   8. Applied rewrites 66.1%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{z + z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z + z}\\ \mathbf{if}\;y \leq -7.4 \cdot 10^{-90} \lor \neg \left(y \leq 7.5 \cdot 10^{+113}\right):\\ \;\;\;\;t\_1 \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(0.5 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z z))))
   (if (or (<= y -7.4e-90) (not (<= y 7.5e+113)))
     (* t_1 (- y))
     (* t_1 (* 0.5 x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + z));
	double tmp;
	if ((y <= -7.4e-90) || !(y <= 7.5e+113)) {
		tmp = t_1 * -y;
	} else {
		tmp = t_1 * (0.5 * x);
	}
	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))
    if ((y <= (-7.4d-90)) .or. (.not. (y <= 7.5d+113))) then
        tmp = t_1 * -y
    else
        tmp = t_1 * (0.5d0 * x)
    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));
	double tmp;
	if ((y <= -7.4e-90) || !(y <= 7.5e+113)) {
		tmp = t_1 * -y;
	} else {
		tmp = t_1 * (0.5 * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z + z))
	tmp = 0
	if (y <= -7.4e-90) or not (y <= 7.5e+113):
		tmp = t_1 * -y
	else:
		tmp = t_1 * (0.5 * x)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z + z))
	tmp = 0.0
	if ((y <= -7.4e-90) || !(y <= 7.5e+113))
		tmp = Float64(t_1 * Float64(-y));
	else
		tmp = Float64(t_1 * Float64(0.5 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + z));
	tmp = 0.0;
	if ((y <= -7.4e-90) || ~((y <= 7.5e+113)))
		tmp = t_1 * -y;
	else
		tmp = t_1 * (0.5 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[y, -7.4e-90], N[Not[LessEqual[y, 7.5e+113]], $MachinePrecision]], N[(t$95$1 * (-y)), $MachinePrecision], N[(t$95$1 * N[(0.5 * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.40000000000000035e-90 or 7.5000000000000001e113 < 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. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   4. 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 68.5

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

   5. Applied rewrites 68.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-1 \cdot \color{blue}{y}\right) \]
   7. 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 60.0

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

   8. Applied rewrites 60.0%

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-y\right) \]
   9. 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 60.0

         \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]

   10. Applied rewrites 60.0%

       \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]

   if -7.40000000000000035e-90 < y < 7.5000000000000001e113

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   4. 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 51.2

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-1 \cdot \color{blue}{y}\right) \]
   7. 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 12.7

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

   8. Applied rewrites 12.7%

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-y\right) \]
   9. 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 12.7

         \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]

   10. Applied rewrites 12.7%

       \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]

   11. Taylor expanded in x around inf

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

       1. lower-*.f64 42.1

          \[\leadsto \sqrt{z + z} \cdot \left(0.5 \cdot x\right) \]

   13. Applied rewrites 42.1%

       \[\leadsto \sqrt{z + z} \cdot \left(0.5 \cdot \color{blue}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{-90} \lor \neg \left(y \leq 7.5 \cdot 10^{+113}\right):\\ \;\;\;\;\sqrt{z + z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z + z} \cdot \left(0.5 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.0% 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 99.9%

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

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
4. 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 58.9

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

5. Applied rewrites 58.9%

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

6. Taylor expanded in x around 0

   \[\leadsto \sqrt{2 \cdot z} \cdot \left(-1 \cdot \color{blue}{y}\right) \]
7. 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 33.8

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

8. Applied rewrites 33.8%

   \[\leadsto \sqrt{2 \cdot z} \cdot \left(-y\right) \]
9. 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 33.8

      \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]

10. Applied rewrites 33.8%

    \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]

11. Taylor expanded in y around 0

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

    1. mul-1-neg N/A

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

    2. lift-neg.f64 N/A

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

    3. +-commutative N/A

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

    4. lift-fma.f64 58.9

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

13. Applied rewrites 58.9%

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

Alternative 12: 30.2% 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 99.9%

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

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
4. 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 58.9

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

5. Applied rewrites 58.9%

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

6. Taylor expanded in x around 0

   \[\leadsto \sqrt{2 \cdot z} \cdot \left(-1 \cdot \color{blue}{y}\right) \]
7. 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 33.8

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

8. Applied rewrites 33.8%

   \[\leadsto \sqrt{2 \cdot z} \cdot \left(-y\right) \]
9. 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 33.8

      \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]

10. Applied rewrites 33.8%

    \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]
11. Add Preprocessing

Developer Target 1: 99.5% 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 2025064 
(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))))