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

Percentage Accurate: 99.4% → 99.4%

Time: 5.8s

Alternatives: 11

Speedup: 1.0×

• 44.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

Alternative 2: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z + z}\\ t_2 := \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\\ \mathbf{if}\;t \leq 0.0255:\\ \;\;\;\;t\_2 \cdot \left(\left(0.5 \cdot x - y\right) \cdot t\_1\right)\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+133}:\\ \;\;\;\;\left(\sqrt{\left(z + z\right) \cdot e^{t \cdot t}} \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \left(x \cdot 0.5 - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z z))) (t_2 (fma (* t t) 0.5 1.0)))
   (if (<= t 0.0255)
     (* t_2 (* (- (* 0.5 x) y) t_1))
     (if (<= t 7.8e+133)
       (* (* (sqrt (* (+ z z) (exp (* t t)))) x) 0.5)
       (* t_1 (* t_2 (- (* x 0.5) y)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + z));
	double t_2 = fma((t * t), 0.5, 1.0);
	double tmp;
	if (t <= 0.0255) {
		tmp = t_2 * (((0.5 * x) - y) * t_1);
	} else if (t <= 7.8e+133) {
		tmp = (sqrt(((z + z) * exp((t * t)))) * x) * 0.5;
	} else {
		tmp = t_1 * (t_2 * ((x * 0.5) - y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z + z))
	t_2 = fma(Float64(t * t), 0.5, 1.0)
	tmp = 0.0
	if (t <= 0.0255)
		tmp = Float64(t_2 * Float64(Float64(Float64(0.5 * x) - y) * t_1));
	elseif (t <= 7.8e+133)
		tmp = Float64(Float64(sqrt(Float64(Float64(z + z) * exp(Float64(t * t)))) * x) * 0.5);
	else
		tmp = Float64(t_1 * Float64(t_2 * Float64(Float64(x * 0.5) - y)));
	end
	return 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[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, If[LessEqual[t, 0.0255], N[(t$95$2 * N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.8e+133], N[(N[(N[Sqrt[N[(N[(z + z), $MachinePrecision] * N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision], N[(t$95$1 * N[(t$95$2 * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 0.0254999999999999984

   1. Initial program 99.4%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 85.6%

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

   if 0.0254999999999999984 < t < 7.80000000000000028e133

   1. Initial program 99.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 62.1%

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

   if 7.80000000000000028e133 < 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

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 85.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. count-2-rev N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 87.4%

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

Alternative 3: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z + z}\\ t_2 := \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\\ \mathbf{if}\;t \leq 0.11:\\ \;\;\;\;t\_2 \cdot \left(\left(0.5 \cdot x - y\right) \cdot t\_1\right)\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+49}:\\ \;\;\;\;\left(-\sqrt{\left(z + z\right) \cdot e^{t \cdot t}}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \left(x \cdot 0.5 - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z z))) (t_2 (fma (* t t) 0.5 1.0)))
   (if (<= t 0.11)
     (* t_2 (* (- (* 0.5 x) y) t_1))
     (if (<= t 1.3e+49)
       (* (- (sqrt (* (+ z z) (exp (* t t))))) y)
       (* t_1 (* t_2 (- (* x 0.5) y)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + z));
	double t_2 = fma((t * t), 0.5, 1.0);
	double tmp;
	if (t <= 0.11) {
		tmp = t_2 * (((0.5 * x) - y) * t_1);
	} else if (t <= 1.3e+49) {
		tmp = -sqrt(((z + z) * exp((t * t)))) * y;
	} else {
		tmp = t_1 * (t_2 * ((x * 0.5) - y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z + z))
	t_2 = fma(Float64(t * t), 0.5, 1.0)
	tmp = 0.0
	if (t <= 0.11)
		tmp = Float64(t_2 * Float64(Float64(Float64(0.5 * x) - y) * t_1));
	elseif (t <= 1.3e+49)
		tmp = Float64(Float64(-sqrt(Float64(Float64(z + z) * exp(Float64(t * t))))) * y);
	else
		tmp = Float64(t_1 * Float64(t_2 * Float64(Float64(x * 0.5) - y)));
	end
	return 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[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, If[LessEqual[t, 0.11], N[(t$95$2 * N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.3e+49], N[((-N[Sqrt[N[(N[(z + z), $MachinePrecision] * N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * y), $MachinePrecision], N[(t$95$1 * N[(t$95$2 * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 0.110000000000000001

   1. Initial program 99.4%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 85.6%

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

   if 0.110000000000000001 < t < 1.29999999999999994e49

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

   4. Applied rewrites 63.7%

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

   if 1.29999999999999994e49 < 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

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 85.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. count-2-rev N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 87.4%

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

Alternative 4: 87.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. distribute-lft1-in N/A

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

   3. +-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. lift--.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-sqrt.f64 N/A

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

4. Applied rewrites 85.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. associate-*r* N/A

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

   8. +-commutative N/A

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

   9. pow2 N/A

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

   10. *-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. count-2-rev N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

6. Applied rewrites 87.4%

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

Alternative 5: 71.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z + z)) * ((x * 0.5d0) - y)
    if (t <= 1.45d0) then
        tmp = t_1
    else
        tmp = ((t * t) * 0.5d0) * t_1
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < 1.44999999999999996

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. metadata-eval N/A

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

      5. pow-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 99.4

          \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \frac{1}{{\color{blue}{\left(z + z\right)}}^{-0.5}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   3. Applied rewrites 99.4%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\frac{1}{{\left(z + z\right)}^{-0.5}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   4. Taylor expanded in t around 0

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

      1. lift-+.f64 N/A

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

      2. pow-flip N/A

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

      3. lift-+.f64 N/A

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

      4. count-2-rev N/A

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

      5. metadata-eval N/A

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

      6. pow1/2 N/A

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

      7. *-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-neg N/A

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

      13. pow1/2 N/A

          \[\leadsto \frac{\frac{1}{2} \cdot x - y}{\frac{1}{\sqrt{2 \cdot z}}} \]

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 N/A

          \[\leadsto \frac{\frac{1}{2} \cdot x - y}{\frac{1}{\sqrt{2 \cdot z}}} \]

      16. count-2-rev N/A

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

      17. lift-+.f64 56.9

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

   6. Applied rewrites 56.9%

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

   7. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 57.0

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

   9. Applied rewrites 57.0%

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

   if 1.44999999999999996 < 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

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 85.6%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 41.5

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

   7. Applied rewrites 41.5%

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

Alternative 6: 66.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z + z}\\ \mathbf{if}\;t \leq 4.4 \cdot 10^{+56}:\\ \;\;\;\;t\_1 \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot t\right) \cdot x\right) \cdot t\_1\right) \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z z))))
   (if (<= t 4.4e+56) (* t_1 (- (* x 0.5) y)) (* (* (* (* t t) x) t_1) 0.25))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + z));
	double tmp;
	if (t <= 4.4e+56) {
		tmp = t_1 * ((x * 0.5) - y);
	} else {
		tmp = (((t * t) * x) * t_1) * 0.25;
	}
	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 <= 4.4d+56) then
        tmp = t_1 * ((x * 0.5d0) - y)
    else
        tmp = (((t * t) * x) * t_1) * 0.25d0
    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 <= 4.4e+56) {
		tmp = t_1 * ((x * 0.5) - y);
	} else {
		tmp = (((t * t) * x) * t_1) * 0.25;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z + z))
	tmp = 0
	if t <= 4.4e+56:
		tmp = t_1 * ((x * 0.5) - y)
	else:
		tmp = (((t * t) * x) * t_1) * 0.25
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z + z))
	tmp = 0.0
	if (t <= 4.4e+56)
		tmp = Float64(t_1 * Float64(Float64(x * 0.5) - y));
	else
		tmp = Float64(Float64(Float64(Float64(t * t) * x) * t_1) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + z));
	tmp = 0.0;
	if (t <= 4.4e+56)
		tmp = t_1 * ((x * 0.5) - y);
	else
		tmp = (((t * t) * x) * t_1) * 0.25;
	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, 4.4e+56], N[(t$95$1 * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t * t), $MachinePrecision] * x), $MachinePrecision] * t$95$1), $MachinePrecision] * 0.25), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 4.40000000000000032e56

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. metadata-eval N/A

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

      5. pow-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 99.4

          \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \frac{1}{{\color{blue}{\left(z + z\right)}}^{-0.5}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   3. Applied rewrites 99.4%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\frac{1}{{\left(z + z\right)}^{-0.5}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   4. Taylor expanded in t around 0

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

      1. lift-+.f64 N/A

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

      2. pow-flip N/A

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

      3. lift-+.f64 N/A

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

      4. count-2-rev N/A

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

      5. metadata-eval N/A

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

      6. pow1/2 N/A

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

      7. *-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-neg N/A

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

      13. pow1/2 N/A

          \[\leadsto \frac{\frac{1}{2} \cdot x - y}{\frac{1}{\sqrt{2 \cdot z}}} \]

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 N/A

          \[\leadsto \frac{\frac{1}{2} \cdot x - y}{\frac{1}{\sqrt{2 \cdot z}}} \]

      16. count-2-rev N/A

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

      17. lift-+.f64 56.9

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

   6. Applied rewrites 56.9%

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

   7. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 57.0

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

   9. Applied rewrites 57.0%

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

   if 4.40000000000000032e56 < 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

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

   4. Applied rewrites 85.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. count-2-rev N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-+.f64 50.9

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

   7. Applied rewrites 50.9%

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

   8. Taylor expanded in t around inf

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

      1. *-commutative N/A

         \[\leadsto \left({t}^{2} \cdot \left(x \cdot \sqrt{2 \cdot z}\right)\right) \cdot \frac{1}{4} \]

      2. lower-*.f64 N/A

         \[\leadsto \left({t}^{2} \cdot \left(x \cdot \sqrt{2 \cdot z}\right)\right) \cdot \frac{1}{4} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left({t}^{2} \cdot x\right) \cdot \sqrt{2 \cdot z}\right) \cdot \frac{1}{4} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left({t}^{2} \cdot x\right) \cdot \sqrt{2 \cdot z}\right) \cdot \frac{1}{4} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left({t}^{2} \cdot x\right) \cdot \sqrt{2 \cdot z}\right) \cdot \frac{1}{4} \]

      6. pow2 N/A

         \[\leadsto \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \sqrt{2 \cdot z}\right) \cdot \frac{1}{4} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \sqrt{2 \cdot z}\right) \cdot \frac{1}{4} \]

      8. count-2-rev N/A

         \[\leadsto \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \sqrt{z + z}\right) \cdot \frac{1}{4} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \sqrt{z + z}\right) \cdot \frac{1}{4} \]

      10. lift-+.f64 28.4

          \[\leadsto \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \sqrt{z + z}\right) \cdot 0.25 \]

   10. Applied rewrites 28.4%

       \[\leadsto \left(\left(\left(t \cdot t\right) \cdot x\right) \cdot \sqrt{z + z}\right) \cdot 0.25 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 57.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{z + z} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+183}:\\ \;\;\;\;\left(-\sqrt{\sqrt{\left(z + z\right) \cdot \left(z + z\right)}}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot z\right) \cdot \sqrt{\frac{2}{z}}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 9.5e+79)
   (* (sqrt (+ z z)) (- (* x 0.5) y))
   (if (<= t 8.2e+183)
     (* (- (sqrt (sqrt (* (+ z z) (+ z z))))) y)
     (* (* (* x z) (sqrt (/ 2.0 z))) 0.5))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 9.5e+79) {
		tmp = sqrt((z + z)) * ((x * 0.5) - y);
	} else if (t <= 8.2e+183) {
		tmp = -sqrt(sqrt(((z + z) * (z + z)))) * y;
	} else {
		tmp = ((x * z) * sqrt((2.0 / z))) * 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) :: tmp
    if (t <= 9.5d+79) then
        tmp = sqrt((z + z)) * ((x * 0.5d0) - y)
    else if (t <= 8.2d+183) then
        tmp = -sqrt(sqrt(((z + z) * (z + z)))) * y
    else
        tmp = ((x * z) * sqrt((2.0d0 / z))) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 9.5e+79) {
		tmp = Math.sqrt((z + z)) * ((x * 0.5) - y);
	} else if (t <= 8.2e+183) {
		tmp = -Math.sqrt(Math.sqrt(((z + z) * (z + z)))) * y;
	} else {
		tmp = ((x * z) * Math.sqrt((2.0 / z))) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 9.5e+79:
		tmp = math.sqrt((z + z)) * ((x * 0.5) - y)
	elif t <= 8.2e+183:
		tmp = -math.sqrt(math.sqrt(((z + z) * (z + z)))) * y
	else:
		tmp = ((x * z) * math.sqrt((2.0 / z))) * 0.5
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 9.5e+79)
		tmp = Float64(sqrt(Float64(z + z)) * Float64(Float64(x * 0.5) - y));
	elseif (t <= 8.2e+183)
		tmp = Float64(Float64(-sqrt(sqrt(Float64(Float64(z + z) * Float64(z + z))))) * y);
	else
		tmp = Float64(Float64(Float64(x * z) * sqrt(Float64(2.0 / z))) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 9.5e+79)
		tmp = sqrt((z + z)) * ((x * 0.5) - y);
	elseif (t <= 8.2e+183)
		tmp = -sqrt(sqrt(((z + z) * (z + z)))) * y;
	else
		tmp = ((x * z) * sqrt((2.0 / z))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 9.5e+79], N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.2e+183], N[((-N[Sqrt[N[Sqrt[N[(N[(z + z), $MachinePrecision] * N[(z + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]) * y), $MachinePrecision], N[(N[(N[(x * z), $MachinePrecision] * N[Sqrt[N[(2.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 9.49999999999999994e79

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. metadata-eval N/A

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

      5. pow-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 99.4

          \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \frac{1}{{\color{blue}{\left(z + z\right)}}^{-0.5}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   3. Applied rewrites 99.4%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\frac{1}{{\left(z + z\right)}^{-0.5}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   4. Taylor expanded in t around 0

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

      1. lift-+.f64 N/A

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

      2. pow-flip N/A

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

      3. lift-+.f64 N/A

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

      4. count-2-rev N/A

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

      5. metadata-eval N/A

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

      6. pow1/2 N/A

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

      7. *-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-neg N/A

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

      13. pow1/2 N/A

          \[\leadsto \frac{\frac{1}{2} \cdot x - y}{\frac{1}{\sqrt{2 \cdot z}}} \]

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 N/A

          \[\leadsto \frac{\frac{1}{2} \cdot x - y}{\frac{1}{\sqrt{2 \cdot z}}} \]

      16. count-2-rev N/A

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

      17. lift-+.f64 56.9

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

   6. Applied rewrites 56.9%

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

   7. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 57.0

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

   9. Applied rewrites 57.0%

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

   if 9.49999999999999994e79 < t < 8.20000000000000029e183

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

   4. Applied rewrites 63.7%

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

   5. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-+.f64 30.4

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

   7. Applied rewrites 30.4%

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

      1. lift-+.f64 N/A

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

      2. rem-square-sqrt N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 28.3

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

   9. Applied rewrites 28.3%

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

   if 8.20000000000000029e183 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. metadata-eval N/A

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

      5. pow-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 99.4

          \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \frac{1}{{\color{blue}{\left(z + z\right)}}^{-0.5}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   3. Applied rewrites 99.4%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\frac{1}{{\left(z + z\right)}^{-0.5}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   4. Taylor expanded in t around 0

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

      1. lift-+.f64 N/A

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

      2. pow-flip N/A

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

      3. lift-+.f64 N/A

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

      4. count-2-rev N/A

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

      5. metadata-eval N/A

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

      6. pow1/2 N/A

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

      7. *-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-neg N/A

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

      13. pow1/2 N/A

          \[\leadsto \frac{\frac{1}{2} \cdot x - y}{\frac{1}{\sqrt{2 \cdot z}}} \]

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 N/A

          \[\leadsto \frac{\frac{1}{2} \cdot x - y}{\frac{1}{\sqrt{2 \cdot z}}} \]

      16. count-2-rev N/A

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

      17. lift-+.f64 56.9

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

   6. Applied rewrites 56.9%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-+.f64 29.2

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

   9. Applied rewrites 29.2%

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

   10. Taylor expanded in z around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-/.f64 29.1

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

   12. Applied rewrites 29.1%

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

Alternative 8: 57.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{+185}:\\ \;\;\;\;\sqrt{z + z} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot z\right) \cdot \sqrt{\frac{2}{z}}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.3e+185)
   (* (sqrt (+ z z)) (- (* x 0.5) y))
   (* (* (* x z) (sqrt (/ 2.0 z))) 0.5)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.3e+185) {
		tmp = sqrt((z + z)) * ((x * 0.5) - y);
	} else {
		tmp = ((x * z) * sqrt((2.0 / z))) * 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) :: tmp
    if (t <= 1.3d+185) then
        tmp = sqrt((z + z)) * ((x * 0.5d0) - y)
    else
        tmp = ((x * z) * sqrt((2.0d0 / z))) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.3e+185) {
		tmp = Math.sqrt((z + z)) * ((x * 0.5) - y);
	} else {
		tmp = ((x * z) * Math.sqrt((2.0 / z))) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 1.3e+185:
		tmp = math.sqrt((z + z)) * ((x * 0.5) - y)
	else:
		tmp = ((x * z) * math.sqrt((2.0 / z))) * 0.5
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.3e+185)
		tmp = Float64(sqrt(Float64(z + z)) * Float64(Float64(x * 0.5) - y));
	else
		tmp = Float64(Float64(Float64(x * z) * sqrt(Float64(2.0 / z))) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 1.3e+185)
		tmp = sqrt((z + z)) * ((x * 0.5) - y);
	else
		tmp = ((x * z) * sqrt((2.0 / z))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.3e+185], N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * z), $MachinePrecision] * N[Sqrt[N[(2.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.3e185

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. metadata-eval N/A

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

      5. pow-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 99.4

          \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \frac{1}{{\color{blue}{\left(z + z\right)}}^{-0.5}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   3. Applied rewrites 99.4%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\frac{1}{{\left(z + z\right)}^{-0.5}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   4. Taylor expanded in t around 0

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

      1. lift-+.f64 N/A

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

      2. pow-flip N/A

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

      3. lift-+.f64 N/A

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

      4. count-2-rev N/A

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

      5. metadata-eval N/A

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

      6. pow1/2 N/A

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

      7. *-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-neg N/A

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

      13. pow1/2 N/A

          \[\leadsto \frac{\frac{1}{2} \cdot x - y}{\frac{1}{\sqrt{2 \cdot z}}} \]

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 N/A

          \[\leadsto \frac{\frac{1}{2} \cdot x - y}{\frac{1}{\sqrt{2 \cdot z}}} \]

      16. count-2-rev N/A

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

      17. lift-+.f64 56.9

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

   6. Applied rewrites 56.9%

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

   7. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 57.0

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

   9. Applied rewrites 57.0%

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

   if 1.3e185 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. metadata-eval N/A

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

      5. pow-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 99.4

          \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \frac{1}{{\color{blue}{\left(z + z\right)}}^{-0.5}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   3. Applied rewrites 99.4%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\frac{1}{{\left(z + z\right)}^{-0.5}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   4. Taylor expanded in t around 0

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

      1. lift-+.f64 N/A

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

      2. pow-flip N/A

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

      3. lift-+.f64 N/A

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

      4. count-2-rev N/A

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

      5. metadata-eval N/A

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

      6. pow1/2 N/A

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

      7. *-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-neg N/A

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

      13. pow1/2 N/A

          \[\leadsto \frac{\frac{1}{2} \cdot x - y}{\frac{1}{\sqrt{2 \cdot z}}} \]

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 N/A

          \[\leadsto \frac{\frac{1}{2} \cdot x - y}{\frac{1}{\sqrt{2 \cdot z}}} \]

      16. count-2-rev N/A

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

      17. lift-+.f64 56.9

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

   6. Applied rewrites 56.9%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-+.f64 29.2

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

   9. Applied rewrites 29.2%

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

   10. Taylor expanded in z around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-/.f64 29.1

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

   12. Applied rewrites 29.1%

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

Alternative 9: 56.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. pow1/2 N/A

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

   4. metadata-eval N/A

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

   5. pow-neg N/A

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

   6. lower-/.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. *-commutative N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f64 99.4

       \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \frac{1}{{\color{blue}{\left(z + z\right)}}^{-0.5}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

3. Applied rewrites 99.4%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\frac{1}{{\left(z + z\right)}^{-0.5}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

4. Taylor expanded in t around 0

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

   1. lift-+.f64 N/A

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

   2. pow-flip N/A

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

   3. lift-+.f64 N/A

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

   4. count-2-rev N/A

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

   5. metadata-eval N/A

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

   6. pow1/2 N/A

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

   7. *-commutative N/A

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

   8. lower-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. pow-neg N/A

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

   13. pow1/2 N/A

       \[\leadsto \frac{\frac{1}{2} \cdot x - y}{\frac{1}{\sqrt{2 \cdot z}}} \]

   14. lower-/.f64 N/A

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

   15. lower-sqrt.f64 N/A

       \[\leadsto \frac{\frac{1}{2} \cdot x - y}{\frac{1}{\sqrt{2 \cdot z}}} \]

   16. count-2-rev N/A

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

   17. lift-+.f64 56.9

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

6. Applied rewrites 56.9%

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

7. Taylor expanded in z around 0

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

   1. lower-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift--.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 57.0

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

9. Applied rewrites 57.0%

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

Alternative 10: 43.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z + z}\\ t_2 := \left(t\_1 \cdot x\right) \cdot 0.5\\ \mathbf{if}\;x \leq -1.08 \cdot 10^{+52}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-56}:\\ \;\;\;\;\left(-t\_1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z z))) (t_2 (* (* t_1 x) 0.5)))
   (if (<= x -1.08e+52) t_2 (if (<= x 3.8e-56) (* (- 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 = (t_1 * x) * 0.5;
	double tmp;
	if (x <= -1.08e+52) {
		tmp = t_2;
	} else if (x <= 3.8e-56) {
		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 = (t_1 * x) * 0.5d0
    if (x <= (-1.08d+52)) then
        tmp = t_2
    else if (x <= 3.8d-56) 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 = (t_1 * x) * 0.5;
	double tmp;
	if (x <= -1.08e+52) {
		tmp = t_2;
	} else if (x <= 3.8e-56) {
		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 = (t_1 * x) * 0.5
	tmp = 0
	if x <= -1.08e+52:
		tmp = t_2
	elif x <= 3.8e-56:
		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(t_1 * x) * 0.5)
	tmp = 0.0
	if (x <= -1.08e+52)
		tmp = t_2;
	elseif (x <= 3.8e-56)
		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 = (t_1 * x) * 0.5;
	tmp = 0.0;
	if (x <= -1.08e+52)
		tmp = t_2;
	elseif (x <= 3.8e-56)
		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[(t$95$1 * x), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x, -1.08e+52], t$95$2, If[LessEqual[x, 3.8e-56], N[((-t$95$1) * y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.07999999999999997e52 or 3.8000000000000002e-56 < x

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. metadata-eval N/A

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

      5. pow-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 99.4

          \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \frac{1}{{\color{blue}{\left(z + z\right)}}^{-0.5}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   3. Applied rewrites 99.4%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\frac{1}{{\left(z + z\right)}^{-0.5}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   4. Taylor expanded in t around 0

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

      1. lift-+.f64 N/A

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

      2. pow-flip N/A

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

      3. lift-+.f64 N/A

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

      4. count-2-rev N/A

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

      5. metadata-eval N/A

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

      6. pow1/2 N/A

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

      7. *-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-neg N/A

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

      13. pow1/2 N/A

          \[\leadsto \frac{\frac{1}{2} \cdot x - y}{\frac{1}{\sqrt{2 \cdot z}}} \]

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 N/A

          \[\leadsto \frac{\frac{1}{2} \cdot x - y}{\frac{1}{\sqrt{2 \cdot z}}} \]

      16. count-2-rev N/A

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

      17. lift-+.f64 56.9

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

   6. Applied rewrites 56.9%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-+.f64 29.2

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

   9. Applied rewrites 29.2%

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

   if -1.07999999999999997e52 < x < 3.8000000000000002e-56

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

   4. Applied rewrites 63.7%

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

   5. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-+.f64 30.4

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

   7. Applied rewrites 30.4%

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

Alternative 11: 30.4% 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 99.4%

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

2. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f64 N/A

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

4. Applied rewrites 63.7%

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

5. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. count-2-rev N/A

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

   5. lift-+.f64 30.4

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

7. Applied rewrites 30.4%

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

Reproduce

herbie shell --seed 2025139 
(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))))