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

Percentage Accurate: 99.3% → 99.3%

Time: 5.1s

Alternatives: 11

Speedup: 1.0×

• 44.7% 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: 99.3% 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: 99.3% 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 99.3%

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

Alternative 2: 88.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\\ t_2 := \sqrt{\left(z + z\right) \cdot e^{t \cdot t}}\\ \mathbf{if}\;t \leq 1.15:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 10^{+91}:\\ \;\;\;\;\left(t\_2 \cdot 0.5\right) \cdot x\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+140}:\\ \;\;\;\;\left(-t\_2\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (fma (* t t) 0.5 1.0)))
        (t_2 (sqrt (* (+ z z) (exp (* t t))))))
   (if (<= t 1.15)
     t_1
     (if (<= t 1e+91)
       (* (* t_2 0.5) x)
       (if (<= t 4e+140) (* (- t_2) y) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (((x * 0.5) - y) * sqrt((z * 2.0))) * fma((t * t), 0.5, 1.0);
	double t_2 = sqrt(((z + z) * exp((t * t))));
	double tmp;
	if (t <= 1.15) {
		tmp = t_1;
	} else if (t <= 1e+91) {
		tmp = (t_2 * 0.5) * x;
	} else if (t <= 4e+140) {
		tmp = -t_2 * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * fma(Float64(t * t), 0.5, 1.0))
	t_2 = sqrt(Float64(Float64(z + z) * exp(Float64(t * t))))
	tmp = 0.0
	if (t <= 1.15)
		tmp = t_1;
	elseif (t <= 1e+91)
		tmp = Float64(Float64(t_2 * 0.5) * x);
	elseif (t <= 4e+140)
		tmp = Float64(Float64(-t_2) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(z + z), $MachinePrecision] * N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 1.15], t$95$1, If[LessEqual[t, 1e+91], N[(N[(t$95$2 * 0.5), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t, 4e+140], N[((-t$95$2) * y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.1499999999999999 or 4.00000000000000024e140 < t

   1. Initial program 99.4%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 90.7

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

   4. Applied rewrites 90.7%

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

   if 1.1499999999999999 < t < 1.00000000000000008e91

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 75.8%

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

   if 1.00000000000000008e91 < t < 4.00000000000000024e140

   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 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 72.4%

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

Alternative 3: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\\ \mathbf{if}\;t \leq 60:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+140}:\\ \;\;\;\;\left(-\sqrt{\left(z + z\right) \cdot e^{t \cdot t}}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (fma (* t t) 0.5 1.0))))
   (if (<= t 60.0)
     t_1
     (if (<= t 4e+140) (* (- (sqrt (* (+ z z) (exp (* t t))))) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (((x * 0.5) - y) * sqrt((z * 2.0))) * fma((t * t), 0.5, 1.0);
	double tmp;
	if (t <= 60.0) {
		tmp = t_1;
	} else if (t <= 4e+140) {
		tmp = -sqrt(((z + z) * exp((t * t)))) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * fma(Float64(t * t), 0.5, 1.0))
	tmp = 0.0
	if (t <= 60.0)
		tmp = t_1;
	elseif (t <= 4e+140)
		tmp = Float64(Float64(-sqrt(Float64(Float64(z + z) * exp(Float64(t * t))))) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 60.0], t$95$1, If[LessEqual[t, 4e+140], N[((-N[Sqrt[N[(N[(z + z), $MachinePrecision] * N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < 60 or 4.00000000000000024e140 < t

   1. Initial program 99.4%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 90.6

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

   4. Applied rewrites 90.6%

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

   if 60 < t < 4.00000000000000024e140

   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 74.3%

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

Alternative 4: 85.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 85.2

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

4. Applied rewrites 85.2%

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

Alternative 5: 65.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z + z}\\ \mathbf{if}\;t \leq 25:\\ \;\;\;\;t\_1 \cdot \left(0.5 \cdot x - y\right)\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+31}:\\ \;\;\;\;\sqrt{\left(z + z\right) \cdot e^{t \cdot t}} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot t\right) \cdot t\_1\right) \cdot -0.5 - t\_1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z z))))
   (if (<= t 25.0)
     (* t_1 (- (* 0.5 x) y))
     (if (<= t 6.5e+31)
       (* (sqrt (* (+ z z) (exp (* t t)))) y)
       (* (- (* (* (* t t) t_1) -0.5) t_1) y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + z));
	double tmp;
	if (t <= 25.0) {
		tmp = t_1 * ((0.5 * x) - y);
	} else if (t <= 6.5e+31) {
		tmp = sqrt(((z + z) * exp((t * t)))) * y;
	} else {
		tmp = ((((t * t) * t_1) * -0.5) - t_1) * y;
	}
	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 <= 25.0d0) then
        tmp = t_1 * ((0.5d0 * x) - y)
    else if (t <= 6.5d+31) then
        tmp = sqrt(((z + z) * exp((t * t)))) * y
    else
        tmp = ((((t * t) * t_1) * (-0.5d0)) - t_1) * y
    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 <= 25.0) {
		tmp = t_1 * ((0.5 * x) - y);
	} else if (t <= 6.5e+31) {
		tmp = Math.sqrt(((z + z) * Math.exp((t * t)))) * y;
	} else {
		tmp = ((((t * t) * t_1) * -0.5) - t_1) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z + z))
	tmp = 0
	if t <= 25.0:
		tmp = t_1 * ((0.5 * x) - y)
	elif t <= 6.5e+31:
		tmp = math.sqrt(((z + z) * math.exp((t * t)))) * y
	else:
		tmp = ((((t * t) * t_1) * -0.5) - t_1) * y
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z + z))
	tmp = 0.0
	if (t <= 25.0)
		tmp = Float64(t_1 * Float64(Float64(0.5 * x) - y));
	elseif (t <= 6.5e+31)
		tmp = Float64(sqrt(Float64(Float64(z + z) * exp(Float64(t * t)))) * y);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(t * t) * t_1) * -0.5) - t_1) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + z));
	tmp = 0.0;
	if (t <= 25.0)
		tmp = t_1 * ((0.5 * x) - y);
	elseif (t <= 6.5e+31)
		tmp = sqrt(((z + z) * exp((t * t)))) * y;
	else
		tmp = ((((t * t) * t_1) * -0.5) - t_1) * y;
	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, 25.0], N[(t$95$1 * N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.5e+31], N[(N[Sqrt[N[(N[(z + z), $MachinePrecision] * N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[(N[(t * t), $MachinePrecision] * t$95$1), $MachinePrecision] * -0.5), $MachinePrecision] - t$95$1), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 25

   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 70.0

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

   4. Applied rewrites 70.0%

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

   if 25 < t < 6.5000000000000004e31

   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 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 75.2%

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

   5. Taylor expanded in z around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-sqrt.f64 24.8

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

   7. Applied rewrites 24.8%

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

   if 6.5000000000000004e31 < t

   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 75.0%

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

   5. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

      10. *-commutative N/A

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

      11. lower-sqrt.f64 N/A

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

      12. count-2-rev N/A

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

      13. lift-+.f64 N/A

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

      14. sqrt-prod N/A

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

      15. *-commutative N/A

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

      16. lower-sqrt.f64 N/A

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

      17. count-2-rev N/A

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

      18. lift-+.f64 56.9

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

   7. Applied rewrites 56.9%

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

Alternative 6: 65.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z + z}\\ \mathbf{if}\;t \leq 1.15 \cdot 10^{+56}:\\ \;\;\;\;t\_1 \cdot \left(0.5 \cdot x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot t\right) \cdot t\_1\right) \cdot -0.5 - t\_1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z z))))
   (if (<= t 1.15e+56)
     (* t_1 (- (* 0.5 x) y))
     (* (- (* (* (* t t) t_1) -0.5) t_1) y))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + z));
	double tmp;
	if (t <= 1.15e+56) {
		tmp = t_1 * ((0.5 * x) - y);
	} else {
		tmp = ((((t * t) * t_1) * -0.5) - t_1) * y;
	}
	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 <= 1.15d+56) then
        tmp = t_1 * ((0.5d0 * x) - y)
    else
        tmp = ((((t * t) * t_1) * (-0.5d0)) - t_1) * y
    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 <= 1.15e+56) {
		tmp = t_1 * ((0.5 * x) - y);
	} else {
		tmp = ((((t * t) * t_1) * -0.5) - t_1) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z + z))
	tmp = 0
	if t <= 1.15e+56:
		tmp = t_1 * ((0.5 * x) - y)
	else:
		tmp = ((((t * t) * t_1) * -0.5) - t_1) * y
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z + z))
	tmp = 0.0
	if (t <= 1.15e+56)
		tmp = Float64(t_1 * Float64(Float64(0.5 * x) - y));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(t * t) * t_1) * -0.5) - t_1) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + z));
	tmp = 0.0;
	if (t <= 1.15e+56)
		tmp = t_1 * ((0.5 * x) - y);
	else
		tmp = ((((t * t) * t_1) * -0.5) - t_1) * y;
	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, 1.15e+56], N[(t$95$1 * N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(t * t), $MachinePrecision] * t$95$1), $MachinePrecision] * -0.5), $MachinePrecision] - t$95$1), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.15000000000000007e56

   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 66.9

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

   4. Applied rewrites 66.9%

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

   if 1.15000000000000007e56 < t

   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 75.1%

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

   5. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

      10. *-commutative N/A

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

      11. lower-sqrt.f64 N/A

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

      12. count-2-rev N/A

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

      13. lift-+.f64 N/A

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

      14. sqrt-prod N/A

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

      15. *-commutative N/A

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

      16. lower-sqrt.f64 N/A

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

      17. count-2-rev N/A

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

      18. lift-+.f64 61.1

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

   7. Applied rewrites 61.1%

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

Alternative 7: 65.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{+31}:\\ \;\;\;\;\sqrt{z + z} \cdot \left(0.5 \cdot x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{\left(z + z\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 5.5e+31)
   (* (sqrt (+ z z)) (- (* 0.5 x) y))
   (* (- (sqrt (* (+ z z) (fma t t 1.0)))) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 5.5e+31) {
		tmp = sqrt((z + z)) * ((0.5 * x) - y);
	} else {
		tmp = -sqrt(((z + z) * fma(t, t, 1.0))) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 5.5e+31)
		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)))) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 5.5e+31], N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(z + z), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.50000000000000002e31

   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 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 5.50000000000000002e31 < t

   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 75.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 55.6

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

   7. Applied rewrites 55.6%

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

Alternative 8: 56.4% 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 99.3%

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

2. Taylor expanded in 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 56.4

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

4. Applied rewrites 56.4%

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

Alternative 9: 42.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z + z}\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{+51}:\\ \;\;\;\;\left(\left(\sqrt{2} \cdot x\right) \cdot 0.5\right) \cdot \sqrt{z}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+109}:\\ \;\;\;\;\left(-t\_1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot t\_1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z z))))
   (if (<= x -1.2e+51)
     (* (* (* (sqrt 2.0) x) 0.5) (sqrt z))
     (if (<= x 8.5e+109) (* (- t_1) y) (* (* x t_1) 0.5)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + z));
	double tmp;
	if (x <= -1.2e+51) {
		tmp = ((sqrt(2.0) * x) * 0.5) * sqrt(z);
	} else if (x <= 8.5e+109) {
		tmp = -t_1 * y;
	} else {
		tmp = (x * t_1) * 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 (x <= (-1.2d+51)) then
        tmp = ((sqrt(2.0d0) * x) * 0.5d0) * sqrt(z)
    else if (x <= 8.5d+109) then
        tmp = -t_1 * y
    else
        tmp = (x * t_1) * 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 (x <= -1.2e+51) {
		tmp = ((Math.sqrt(2.0) * x) * 0.5) * Math.sqrt(z);
	} else if (x <= 8.5e+109) {
		tmp = -t_1 * y;
	} else {
		tmp = (x * t_1) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z + z))
	tmp = 0
	if x <= -1.2e+51:
		tmp = ((math.sqrt(2.0) * x) * 0.5) * math.sqrt(z)
	elif x <= 8.5e+109:
		tmp = -t_1 * y
	else:
		tmp = (x * t_1) * 0.5
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z + z))
	tmp = 0.0
	if (x <= -1.2e+51)
		tmp = Float64(Float64(Float64(sqrt(2.0) * x) * 0.5) * sqrt(z));
	elseif (x <= 8.5e+109)
		tmp = Float64(Float64(-t_1) * y);
	else
		tmp = Float64(Float64(x * t_1) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + z));
	tmp = 0.0;
	if (x <= -1.2e+51)
		tmp = ((sqrt(2.0) * x) * 0.5) * sqrt(z);
	elseif (x <= 8.5e+109)
		tmp = -t_1 * y;
	else
		tmp = (x * t_1) * 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[x, -1.2e+51], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.5e+109], N[((-t$95$1) * y), $MachinePrecision], N[(N[(x * t$95$1), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1999999999999999e51

   1. Initial program 99.8%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in t around 0

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 59.6

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

   6. Applied rewrites 59.6%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 49.5

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

   9. Applied rewrites 49.5%

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

   if -1.1999999999999999e51 < x < 8.5000000000000004e109

   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 75.9%

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

   5. Taylor expanded in t around 0

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

      1. count-2-rev N/A

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

      2. lift-+.f64 38.1

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

   7. Applied rewrites 38.1%

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

   if 8.5000000000000004e109 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. sqrt-prod N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. 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 62.5

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in x around inf

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

      1. lift-+.f64 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. sqrt-prod N/A

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

      14. *-commutative N/A

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

      15. lower-sqrt.f64 N/A

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

      16. count-2-rev N/A

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

      17. lift-+.f64 52.7

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

   7. Applied rewrites 52.7%

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

Alternative 10: 42.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z + z}\\ t_2 := \left(x \cdot t\_1\right) \cdot 0.5\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{+51}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+109}:\\ \;\;\;\;\left(-t\_1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z z))) (t_2 (* (* x t_1) 0.5)))
   (if (<= x -1.2e+51) t_2 (if (<= x 8.5e+109) (* (- t_1) y) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + z));
	double t_2 = (x * t_1) * 0.5;
	double tmp;
	if (x <= -1.2e+51) {
		tmp = t_2;
	} else if (x <= 8.5e+109) {
		tmp = -t_1 * y;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = sqrt((z + z))
    t_2 = (x * t_1) * 0.5d0
    if (x <= (-1.2d+51)) then
        tmp = t_2
    else if (x <= 8.5d+109) then
        tmp = -t_1 * y
    else
        tmp = t_2
    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 t_2 = (x * t_1) * 0.5;
	double tmp;
	if (x <= -1.2e+51) {
		tmp = t_2;
	} else if (x <= 8.5e+109) {
		tmp = -t_1 * y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z + z))
	t_2 = (x * t_1) * 0.5
	tmp = 0
	if x <= -1.2e+51:
		tmp = t_2
	elif x <= 8.5e+109:
		tmp = -t_1 * y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z + z))
	t_2 = Float64(Float64(x * t_1) * 0.5)
	tmp = 0.0
	if (x <= -1.2e+51)
		tmp = t_2;
	elseif (x <= 8.5e+109)
		tmp = Float64(Float64(-t_1) * y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + z));
	t_2 = (x * t_1) * 0.5;
	tmp = 0.0;
	if (x <= -1.2e+51)
		tmp = t_2;
	elseif (x <= 8.5e+109)
		tmp = -t_1 * y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * t$95$1), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x, -1.2e+51], t$95$2, If[LessEqual[x, 8.5e+109], N[((-t$95$1) * y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1999999999999999e51 or 8.5000000000000004e109 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. sqrt-prod N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. 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 61.0

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

   4. Applied rewrites 61.0%

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

   5. Taylor expanded in x around inf

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

      1. lift-+.f64 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. sqrt-prod N/A

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

      14. *-commutative N/A

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

      15. lower-sqrt.f64 N/A

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

      16. count-2-rev N/A

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

      17. lift-+.f64 51.0

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

   7. Applied rewrites 51.0%

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

   if -1.1999999999999999e51 < x < 8.5000000000000004e109

   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 75.9%

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

   5. Taylor expanded in t around 0

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

      1. count-2-rev N/A

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

      2. lift-+.f64 38.1

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

   7. Applied rewrites 38.1%

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

Alternative 11: 29.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

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 62.9%

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

5. Taylor expanded in t around 0

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

   1. count-2-rev N/A

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

   2. lift-+.f64 29.2

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

7. Applied rewrites 29.2%

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

Reproduce

herbie shell --seed 2025114 
(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))))