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

Percentage Accurate: 99.4% → 99.4%

Time: 5.1s

Alternatives: 14

Speedup: 0.7×

• 44.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

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(0.5d0) ** (t * t))
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(0.5), (t * t));
}

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * (exp(0.5) ^ (t * t));
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[0.5], $MachinePrecision], N[(t * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. pow2 N/A

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

   5. exp-sqrt N/A

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

   6. pow1/2 N/A

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

   7. exp-prod N/A

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

   8. *-commutative N/A

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

   9. exp-prod N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-exp.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 99.4

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

3. Applied rewrites 99.4%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

Alternative 3: 93.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.00195:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z + z} \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.5 - \frac{y}{x}\right) \cdot x\right) \cdot \sqrt{e^{t \cdot t} \cdot \left(z + z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 0.00195)
   (* (- (* x 0.5) y) (* (sqrt (+ z z)) (fma (* t t) 0.5 1.0)))
   (* (* (- 0.5 (/ y x)) x) (sqrt (* (exp (* t t)) (+ z z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 0.00195) {
		tmp = ((x * 0.5) - y) * (sqrt((z + z)) * fma((t * t), 0.5, 1.0));
	} else {
		tmp = ((0.5 - (y / x)) * x) * sqrt((exp((t * t)) * (z + z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 0.00195)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * Float64(sqrt(Float64(z + z)) * fma(Float64(t * t), 0.5, 1.0)));
	else
		tmp = Float64(Float64(Float64(0.5 - Float64(y / x)) * x) * sqrt(Float64(exp(Float64(t * t)) * Float64(z + z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 0.00195], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 - N[(y / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[Sqrt[N[(N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision] * N[(z + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.0019499999999999999

   1. Initial program 99.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. exp-sqrt N/A

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

      6. pow1/2 N/A

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

      7. exp-prod N/A

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

      8. *-commutative N/A

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

      9. exp-prod N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 99.5

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in t around 0

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)} \]
   5. 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 90.5

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

   6. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 90.9

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f64 90.9

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

   8. Applied rewrites 90.9%

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

   if 0.0019499999999999999 < t

   1. Initial program 99.1%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 99.0

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

   4. Applied rewrites 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. count-2-rev N/A

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

      11. pow2 N/A

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

      12. exp-sqrt-rev N/A

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

      13. pow2 N/A

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

      14. sqrt-prod N/A

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

   6. Applied rewrites 99.7%

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

Alternative 4: 90.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{\left(z + z\right) \cdot e^{t \cdot t}}\\ t_3 := \sqrt{z + z}\\ \mathbf{if}\;t \leq 0.03:\\ \;\;\;\;t\_1 \cdot \left(t\_3 \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\right)\\ \mathbf{elif}\;t \leq 10^{+57}:\\ \;\;\;\;\left(t\_2 \cdot 0.5\right) \cdot x\\ \mathbf{elif}\;t \leq 1.82 \cdot 10^{+132}:\\ \;\;\;\;\left(-t\_2\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(t\_3 \cdot \left(\left(t \cdot t\right) \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y))
        (t_2 (sqrt (* (+ z z) (exp (* t t)))))
        (t_3 (sqrt (+ z z))))
   (if (<= t 0.03)
     (* t_1 (* t_3 (fma (* t t) 0.5 1.0)))
     (if (<= t 1e+57)
       (* (* t_2 0.5) x)
       (if (<= t 1.82e+132) (* (- t_2) y) (* t_1 (* t_3 (* (* t t) 0.5))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = sqrt(((z + z) * exp((t * t))));
	double t_3 = sqrt((z + z));
	double tmp;
	if (t <= 0.03) {
		tmp = t_1 * (t_3 * fma((t * t), 0.5, 1.0));
	} else if (t <= 1e+57) {
		tmp = (t_2 * 0.5) * x;
	} else if (t <= 1.82e+132) {
		tmp = -t_2 * y;
	} else {
		tmp = t_1 * (t_3 * ((t * t) * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = sqrt(Float64(Float64(z + z) * exp(Float64(t * t))))
	t_3 = sqrt(Float64(z + z))
	tmp = 0.0
	if (t <= 0.03)
		tmp = Float64(t_1 * Float64(t_3 * fma(Float64(t * t), 0.5, 1.0)));
	elseif (t <= 1e+57)
		tmp = Float64(Float64(t_2 * 0.5) * x);
	elseif (t <= 1.82e+132)
		tmp = Float64(Float64(-t_2) * y);
	else
		tmp = Float64(t_1 * Float64(t_3 * Float64(Float64(t * t) * 0.5)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(z + z), $MachinePrecision] * N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 0.03], N[(t$95$1 * N[(t$95$3 * N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e+57], N[(N[(t$95$2 * 0.5), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t, 1.82e+132], N[((-t$95$2) * y), $MachinePrecision], N[(t$95$1 * N[(t$95$3 * N[(N[(t * t), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 0.029999999999999999

   1. Initial program 99.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. exp-sqrt N/A

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

      6. pow1/2 N/A

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

      7. exp-prod N/A

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

      8. *-commutative N/A

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

      9. exp-prod N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 99.5

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in t around 0

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)} \]
   5. 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 90.5

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

   6. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 90.9

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f64 90.9

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

   8. Applied rewrites 90.9%

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

   if 0.029999999999999999 < t < 1.00000000000000005e57

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 73.1%

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

   if 1.00000000000000005e57 < t < 1.82e132

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0

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

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

   4. Applied rewrites 72.9%

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

   if 1.82e132 < t

   1. Initial program 98.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. exp-sqrt N/A

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

      6. pow1/2 N/A

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

      7. exp-prod N/A

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

      8. *-commutative N/A

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

      9. exp-prod N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 98.9

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in t around 0

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)} \]
   5. 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 96.1

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

   6. Applied rewrites 96.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 97.5

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f64 97.5

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

   8. Applied rewrites 97.5%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 97.5

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

   11. Applied rewrites 97.5%

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

Alternative 5: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 1.75:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\\ \mathbf{elif}\;t \leq 1.82 \cdot 10^{+132}:\\ \;\;\;\;\left(-\sqrt{\left(z + z\right) \cdot e^{t \cdot t}}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{z + z} \cdot \left(\left(t \cdot t\right) \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 1.75)
     (* (* t_1 (sqrt (* z 2.0))) (fma (* t t) 0.5 1.0))
     (if (<= t 1.82e+132)
       (* (- (sqrt (* (+ z z) (exp (* t t))))) y)
       (* t_1 (* (sqrt (+ z z)) (* (* t t) 0.5)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 1.75) {
		tmp = (t_1 * sqrt((z * 2.0))) * fma((t * t), 0.5, 1.0);
	} else if (t <= 1.82e+132) {
		tmp = -sqrt(((z + z) * exp((t * t)))) * y;
	} else {
		tmp = t_1 * (sqrt((z + z)) * ((t * t) * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 1.75)
		tmp = Float64(Float64(t_1 * sqrt(Float64(z * 2.0))) * fma(Float64(t * t), 0.5, 1.0));
	elseif (t <= 1.82e+132)
		tmp = Float64(Float64(-sqrt(Float64(Float64(z + z) * exp(Float64(t * t))))) * y);
	else
		tmp = Float64(t_1 * Float64(sqrt(Float64(z + z)) * Float64(Float64(t * t) * 0.5)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 1.75], N[(N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.82e+132], N[((-N[Sqrt[N[(N[(z + z), $MachinePrecision] * N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * y), $MachinePrecision], N[(t$95$1 * N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.75

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 90.3

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

   4. Applied rewrites 90.3%

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

   if 1.75 < t < 1.82e132

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

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

   4. Applied rewrites 73.6%

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

   if 1.82e132 < t

   1. Initial program 98.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. exp-sqrt N/A

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

      6. pow1/2 N/A

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

      7. exp-prod N/A

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

      8. *-commutative N/A

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

      9. exp-prod N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 98.9

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in t around 0

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)} \]
   5. 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 96.1

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

   6. Applied rewrites 96.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 97.5

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f64 97.5

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

   8. Applied rewrites 97.5%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 97.5

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

   11. Applied rewrites 97.5%

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

Alternative 6: 87.0% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (- (* x 0.5) y) (* (sqrt (+ z z)) (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 + z)) * fma((t * t), 0.5, 1.0));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-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. exp-prod N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-exp.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 99.4

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in t around 0

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)} \]
5. 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 86.4

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

6. Applied rewrites 86.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-*.f64 87.0

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. count-2-rev N/A

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

   15. lift-+.f64 87.0

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

8. Applied rewrites 87.0%

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

Alternative 7: 72.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z + z}\\ \mathbf{if}\;t \leq 1.45:\\ \;\;\;\;t\_1 \cdot \left(0.5 \cdot x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(t\_1 \cdot \left(\left(t \cdot t\right) \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z z))))
   (if (<= t 1.45)
     (* t_1 (- (* 0.5 x) y))
     (* (- (* x 0.5) y) (* t_1 (* (* t t) 0.5))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + z));
	double tmp;
	if (t <= 1.45) {
		tmp = t_1 * ((0.5 * x) - y);
	} else {
		tmp = ((x * 0.5) - y) * (t_1 * ((t * t) * 0.5));
	}
	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 (t <= 1.45d0) then
        tmp = t_1 * ((0.5d0 * x) - y)
    else
        tmp = ((x * 0.5d0) - y) * (t_1 * ((t * t) * 0.5d0))
    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 (t <= 1.45) {
		tmp = t_1 * ((0.5 * x) - y);
	} else {
		tmp = ((x * 0.5) - y) * (t_1 * ((t * t) * 0.5));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z + z))
	tmp = 0
	if t <= 1.45:
		tmp = t_1 * ((0.5 * x) - y)
	else:
		tmp = ((x * 0.5) - y) * (t_1 * ((t * t) * 0.5))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z + z))
	tmp = 0.0
	if (t <= 1.45)
		tmp = Float64(t_1 * Float64(Float64(0.5 * x) - y));
	else
		tmp = Float64(Float64(Float64(x * 0.5) - y) * Float64(t_1 * Float64(Float64(t * t) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + z));
	tmp = 0.0;
	if (t <= 1.45)
		tmp = t_1 * ((0.5 * x) - y);
	else
		tmp = ((x * 0.5) - y) * (t_1 * ((t * t) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 1.45], N[(t$95$1 * N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[(t$95$1 * N[(N[(t * t), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.44999999999999996

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. sqrt-prod N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lift--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 71.1

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

   4. Applied rewrites 71.1%

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

   if 1.44999999999999996 < t

   1. Initial program 99.1%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. 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. exp-prod N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 99.1

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in t around 0

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)} \]
   5. 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 73.9

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

   6. Applied rewrites 73.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 75.3

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f64 75.3

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

   8. Applied rewrites 75.3%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 75.3

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

   11. Applied rewrites 75.3%

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

Alternative 8: 66.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z + z}\\ \mathbf{if}\;t \leq 1.75 \cdot 10^{+56}:\\ \;\;\;\;t\_1 \cdot \left(0.5 \cdot x - y\right)\\ \mathbf{elif}\;t \leq 10^{+165}:\\ \;\;\;\;\left(-y\right) \cdot \left(t\_1 \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(z + z\right) \cdot \left(t \cdot t\right)} \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z z))))
   (if (<= t 1.75e+56)
     (* t_1 (- (* 0.5 x) y))
     (if (<= t 1e+165)
       (* (- y) (* t_1 (fma (* t t) 0.5 1.0)))
       (* (* (sqrt (* (+ z z) (* t t))) 0.5) x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + z));
	double tmp;
	if (t <= 1.75e+56) {
		tmp = t_1 * ((0.5 * x) - y);
	} else if (t <= 1e+165) {
		tmp = -y * (t_1 * fma((t * t), 0.5, 1.0));
	} else {
		tmp = (sqrt(((z + z) * (t * t))) * 0.5) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z + z))
	tmp = 0.0
	if (t <= 1.75e+56)
		tmp = Float64(t_1 * Float64(Float64(0.5 * x) - y));
	elseif (t <= 1e+165)
		tmp = Float64(Float64(-y) * Float64(t_1 * fma(Float64(t * t), 0.5, 1.0)));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(z + z) * Float64(t * t))) * 0.5) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 1.75e+56], N[(t$95$1 * N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e+165], N[((-y) * N[(t$95$1 * N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(z + z), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.75e56

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. sqrt-prod N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lift--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 68.1

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

   4. Applied rewrites 68.1%

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

   if 1.75e56 < t < 9.99999999999999899e164

   1. Initial program 99.4%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. exp-sqrt N/A

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

      6. pow1/2 N/A

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

      7. exp-prod N/A

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

      8. *-commutative N/A

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

      9. exp-prod N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 99.4

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in t around 0

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)} \]
   5. 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 64.4

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

   6. Applied rewrites 64.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 66.7

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f64 66.7

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

   8. Applied rewrites 66.7%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 43.1

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

   11. Applied rewrites 43.1%

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

   if 9.99999999999999899e164 < t

   1. Initial program 98.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 74.2%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 74.2

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

   7. Applied rewrites 74.2%

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

   8. Taylor expanded in t around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 74.2

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

   10. Applied rewrites 74.2%

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

Alternative 9: 66.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.03:\\ \;\;\;\;\sqrt{z + z} \cdot \left(0.5 \cdot x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(z + z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 0.03)
   (* (sqrt (+ z z)) (- (* 0.5 x) y))
   (* (* (sqrt (* (+ z z) (fma t t 1.0))) 0.5) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 0.03) {
		tmp = sqrt((z + z)) * ((0.5 * x) - y);
	} else {
		tmp = (sqrt(((z + z) * fma(t, t, 1.0))) * 0.5) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 0.03)
		tmp = Float64(sqrt(Float64(z + z)) * Float64(Float64(0.5 * x) - y));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(z + z) * fma(t, t, 1.0))) * 0.5) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 0.03], N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(z + z), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.029999999999999999

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. sqrt-prod N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lift--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 71.2

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

   4. Applied rewrites 71.2%

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

   if 0.029999999999999999 < t

   1. Initial program 99.1%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 74.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 51.7

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

   7. Applied rewrites 51.7%

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

Alternative 10: 66.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{z + z} \cdot \left(0.5 \cdot x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(z + z\right) \cdot \left(t \cdot t\right)} \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 2.4e+52)
   (* (sqrt (+ z z)) (- (* 0.5 x) y))
   (* (* (sqrt (* (+ z z) (* t t))) 0.5) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.4e+52) {
		tmp = sqrt((z + z)) * ((0.5 * x) - y);
	} else {
		tmp = (sqrt(((z + z) * (t * t))) * 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) :: tmp
    if (t <= 2.4d+52) then
        tmp = sqrt((z + z)) * ((0.5d0 * x) - y)
    else
        tmp = (sqrt(((z + z) * (t * t))) * 0.5d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.4e+52) {
		tmp = Math.sqrt((z + z)) * ((0.5 * x) - y);
	} else {
		tmp = (Math.sqrt(((z + z) * (t * t))) * 0.5) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 2.4e+52:
		tmp = math.sqrt((z + z)) * ((0.5 * x) - y)
	else:
		tmp = (math.sqrt(((z + z) * (t * t))) * 0.5) * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 2.4e+52)
		tmp = Float64(sqrt(Float64(z + z)) * Float64(Float64(0.5 * x) - y));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(z + z) * Float64(t * t))) * 0.5) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 2.4e+52)
		tmp = sqrt((z + z)) * ((0.5 * x) - y);
	else
		tmp = (sqrt(((z + z) * (t * t))) * 0.5) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 2.4e+52], N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(z + z), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.4e52

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. sqrt-prod N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lift--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 68.2

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

   4. Applied rewrites 68.2%

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

   if 2.4e52 < t

   1. Initial program 99.0%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 74.2%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 59.2

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

   7. Applied rewrites 59.2%

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

   8. Taylor expanded in t around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 59.2

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

   10. Applied rewrites 59.2%

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

Alternative 11: 57.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return sqrt((z + z)) * ((0.5 * x) - 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)) * ((0.5d0 * x) - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. sqrt-prod N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f64 N/A

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

   11. lift--.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 57.6

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

4. Applied rewrites 57.6%

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

Alternative 12: 43.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z} \cdot \left(\left(-y\right) \cdot \sqrt{2}\right)\\ \mathbf{if}\;y \leq -5.4 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+115}:\\ \;\;\;\;\left(x \cdot \sqrt{z + z}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (sqrt z) (* (- y) (sqrt 2.0)))))
   (if (<= y -5.4e-18)
     t_1
     (if (<= y 4.8e+115) (* (* x (sqrt (+ z z))) 0.5) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt(z) * (-y * sqrt(2.0));
	double tmp;
	if (y <= -5.4e-18) {
		tmp = t_1;
	} else if (y <= 4.8e+115) {
		tmp = (x * sqrt((z + z))) * 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(z) * (-y * sqrt(2.0d0))
    if (y <= (-5.4d-18)) then
        tmp = t_1
    else if (y <= 4.8d+115) then
        tmp = (x * sqrt((z + z))) * 0.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt(z) * (-y * Math.sqrt(2.0));
	double tmp;
	if (y <= -5.4e-18) {
		tmp = t_1;
	} else if (y <= 4.8e+115) {
		tmp = (x * Math.sqrt((z + z))) * 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt(z) * (-y * math.sqrt(2.0))
	tmp = 0
	if y <= -5.4e-18:
		tmp = t_1
	elif y <= 4.8e+115:
		tmp = (x * math.sqrt((z + z))) * 0.5
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(sqrt(z) * Float64(Float64(-y) * sqrt(2.0)))
	tmp = 0.0
	if (y <= -5.4e-18)
		tmp = t_1;
	elseif (y <= 4.8e+115)
		tmp = Float64(Float64(x * sqrt(Float64(z + z))) * 0.5);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt(z) * (-y * sqrt(2.0));
	tmp = 0.0;
	if (y <= -5.4e-18)
		tmp = t_1;
	elseif (y <= 4.8e+115)
		tmp = (x * sqrt((z + z))) * 0.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[z], $MachinePrecision] * N[((-y) * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.4e-18], t$95$1, If[LessEqual[y, 4.8e+115], N[(N[(x * N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.39999999999999977e-18 or 4.8000000000000001e115 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. pow2 N/A

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

      11. exp-sqrt N/A

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

      12. pow1/2 N/A

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

      13. exp-prod N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. sqrt-prod N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-sqrt.f64 83.6

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

   6. Applied rewrites 83.6%

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

   7. Taylor expanded in t around 0

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

      1. lift-sqrt.f64 48.6

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

   9. Applied rewrites 48.6%

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

   if -5.39999999999999977e-18 < y < 4.8000000000000001e115

   1. Initial program 99.1%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 77.3%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-prod N/A

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

      7. *-commutative N/A

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

      8. lower-sqrt.f64 N/A

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

      9. count-2-rev N/A

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

      10. lift-+.f64 40.2

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

   7. Applied rewrites 40.2%

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

Alternative 13: 29.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \sqrt{z + z}\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (* x (sqrt (+ z z))) 0.5))

C

double code(double x, double y, double z, double t) {
	return (x * sqrt((z + z))) * 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 * sqrt((z + z))) * 0.5d0
end function

Java

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

Python

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

Julia

function code(x, y, z, t)
	return Float64(Float64(x * sqrt(Float64(z + z))) * 0.5)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 62.4%

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

5. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. sqrt-prod N/A

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

   7. *-commutative N/A

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

   8. lower-sqrt.f64 N/A

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

   9. count-2-rev N/A

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

   10. lift-+.f64 29.9

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

7. Applied rewrites 29.9%

   \[\leadsto \left(x \cdot \sqrt{z + z}\right) \cdot \color{blue}{0.5} \]
8. Add Preprocessing

Alternative 14: 2.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \left(\sqrt{z + z} \cdot x\right) \end{array} \]

FPCore

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

C

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

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 = (-0.5d0) * (sqrt((z + z)) * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 62.4%

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

5. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. sqrt-prod N/A

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

   7. *-commutative N/A

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

   8. lower-sqrt.f64 N/A

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

   9. count-2-rev N/A

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

   10. lift-+.f64 29.9

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

7. Applied rewrites 29.9%

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

8. Taylor expanded in z around -inf

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. lower-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. sqrt-prod N/A

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

   7. *-commutative N/A

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

   8. count-2-rev N/A

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

   9. *-commutative N/A

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

   10. lift-sqrt.f64 N/A

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

   11. count-2-rev N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lower-*.f64 2.6

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. count-2-rev N/A

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

   18. lift-+.f64 2.6

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

10. Applied rewrites 2.6%

    \[\leadsto -0.5 \cdot \left(\sqrt{z + z} \cdot \color{blue}{x}\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025112 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64
  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))