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

Percentage Accurate: 99.5% → 99.8%

Time: 11.2s

Alternatives: 15

Speedup: 1.1×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (pow (sqrt (exp t_m)) t_m)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sqrt[N[Exp[t$95$m], $MachinePrecision]], $MachinePrecision], t$95$m], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

Alternative 2: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{{\left(e^{t\_m}\right)}^{t\_m}}\right) \cdot \sqrt{z + z} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (pow (exp t_m) t_m))) (sqrt (+ z z))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	return (((x * 0.5) - y) * sqrt(pow(exp(t_m), t_m))) * sqrt((z + z));
}

Fortran

t_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    code = (((x * 0.5d0) - y) * sqrt((exp(t_m) ** t_m))) * sqrt((z + z))
end function

Java

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

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	return (((x * 0.5) - y) * math.sqrt(math.pow(math.exp(t_m), t_m))) * math.sqrt((z + z))

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt((exp(t_m) ^ t_m))) * sqrt(Float64(z + z)))
end

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[Power[N[Exp[t$95$m], $MachinePrecision], t$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{e^{t\_m \cdot t\_m}}\right) \cdot \sqrt{z + z} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (exp (* t_m t_m)))) (sqrt (+ z z))))

C

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

Fortran

t_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    code = (((x * 0.5d0) - y) * sqrt(exp((t_m * t_m)))) * sqrt((z + z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[Exp[N[(t$95$m * t$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.8%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \left(x \cdot 0.5 - y\right) \cdot \sqrt{e^{t\_m \cdot t\_m} \cdot \left(z + z\right)} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (* (- (* x 0.5) y) (sqrt (* (exp (* t_m t_m)) (+ z z)))))

C

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

Fortran

t_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    code = ((x * 0.5d0) - y) * sqrt((exp((t_m * t_m)) * (z + z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(N[Exp[N[(t$95$m * t$95$m), $MachinePrecision]], $MachinePrecision] * N[(z + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.8%

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

Alternative 5: 87.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := \sqrt{z + z}\\ \mathbf{if}\;t\_m \leq 0.00014:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot t\_1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-1 \cdot y\right) \cdot \sqrt{e^{t\_m \cdot t\_m}}\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (sqrt (+ z z))))
   (if (<= t_m 0.00014)
     (* (* (- (* x 0.5) y) t_1) 1.0)
     (* (* (* -1.0 y) (sqrt (exp (* t_m t_m)))) t_1))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = sqrt((z + z));
	double tmp;
	if (t_m <= 0.00014) {
		tmp = (((x * 0.5) - y) * t_1) * 1.0;
	} else {
		tmp = ((-1.0 * y) * sqrt(exp((t_m * t_m)))) * t_1;
	}
	return tmp;
}

Fortran

t_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z + z))
    if (t_m <= 0.00014d0) then
        tmp = (((x * 0.5d0) - y) * t_1) * 1.0d0
    else
        tmp = (((-1.0d0) * y) * sqrt(exp((t_m * t_m)))) * t_1
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = Math.sqrt((z + z));
	double tmp;
	if (t_m <= 0.00014) {
		tmp = (((x * 0.5) - y) * t_1) * 1.0;
	} else {
		tmp = ((-1.0 * y) * Math.sqrt(Math.exp((t_m * t_m)))) * t_1;
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = math.sqrt((z + z))
	tmp = 0
	if t_m <= 0.00014:
		tmp = (((x * 0.5) - y) * t_1) * 1.0
	else:
		tmp = ((-1.0 * y) * math.sqrt(math.exp((t_m * t_m)))) * t_1
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = sqrt(Float64(z + z))
	tmp = 0.0
	if (t_m <= 0.00014)
		tmp = Float64(Float64(Float64(Float64(x * 0.5) - y) * t_1) * 1.0);
	else
		tmp = Float64(Float64(Float64(-1.0 * y) * sqrt(exp(Float64(t_m * t_m)))) * t_1);
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = sqrt((z + z));
	tmp = 0.0;
	if (t_m <= 0.00014)
		tmp = (((x * 0.5) - y) * t_1) * 1.0;
	else
		tmp = ((-1.0 * y) * sqrt(exp((t_m * t_m)))) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$m, 0.00014], N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(-1.0 * y), $MachinePrecision] * N[Sqrt[N[Exp[N[(t$95$m * t$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.3999999999999999e-4

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 57.6%

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

   if 1.3999999999999999e-4 < t

   1. Initial program 99.5%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 63.1%

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

Alternative 6: 87.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 0.00014:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z + z}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot y\right) \cdot \sqrt{e^{t\_m \cdot t\_m} \cdot \left(z + z\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (if (<= t_m 0.00014)
   (* (* (- (* x 0.5) y) (sqrt (+ z z))) 1.0)
   (* (* -1.0 y) (sqrt (* (exp (* t_m t_m)) (+ z z))))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 0.00014) {
		tmp = (((x * 0.5) - y) * sqrt((z + z))) * 1.0;
	} else {
		tmp = (-1.0 * y) * sqrt((exp((t_m * t_m)) * (z + z)));
	}
	return tmp;
}

Fortran

t_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 0.00014d0) then
        tmp = (((x * 0.5d0) - y) * sqrt((z + z))) * 1.0d0
    else
        tmp = ((-1.0d0) * y) * sqrt((exp((t_m * t_m)) * (z + z)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 0.00014) {
		tmp = (((x * 0.5) - y) * Math.sqrt((z + z))) * 1.0;
	} else {
		tmp = (-1.0 * y) * Math.sqrt((Math.exp((t_m * t_m)) * (z + z)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	tmp = 0
	if t_m <= 0.00014:
		tmp = (((x * 0.5) - y) * math.sqrt((z + z))) * 1.0
	else:
		tmp = (-1.0 * y) * math.sqrt((math.exp((t_m * t_m)) * (z + z)))
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	tmp = 0.0
	if (t_m <= 0.00014)
		tmp = Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z + z))) * 1.0);
	else
		tmp = Float64(Float64(-1.0 * y) * sqrt(Float64(exp(Float64(t_m * t_m)) * Float64(z + z))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	tmp = 0.0;
	if (t_m <= 0.00014)
		tmp = (((x * 0.5) - y) * sqrt((z + z))) * 1.0;
	else
		tmp = (-1.0 * y) * sqrt((exp((t_m * t_m)) * (z + z)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := If[LessEqual[t$95$m, 0.00014], N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(-1.0 * y), $MachinePrecision] * N[Sqrt[N[(N[Exp[N[(t$95$m * t$95$m), $MachinePrecision]], $MachinePrecision] * N[(z + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.3999999999999999e-4

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 57.6%

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

   if 1.3999999999999999e-4 < t

   1. Initial program 99.5%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 63.1%

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

Alternative 7: 65.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \left(t\_1 \cdot \sqrt{\sqrt{\left(z + z\right) \cdot \left(z + z\right)}}\right) \cdot 1\\ \mathbf{if}\;t\_m \leq 0.28:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{z + z}\right) \cdot 1\\ \mathbf{elif}\;t\_m \leq 4.8 \cdot 10^{+56}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+118}:\\ \;\;\;\;\left(t\_1 \cdot \left(z \cdot \sqrt{\frac{z + z}{z \cdot z}}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y))
        (t_2 (* (* t_1 (sqrt (sqrt (* (+ z z) (+ z z))))) 1.0)))
   (if (<= t_m 0.28)
     (* (* t_1 (sqrt (+ z z))) 1.0)
     (if (<= t_m 4.8e+56)
       t_2
       (if (<= t_m 2e+118)
         (* (* t_1 (* z (sqrt (/ (+ z z) (* z z))))) 1.0)
         t_2)))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double t_2 = (t_1 * sqrt(sqrt(((z + z) * (z + z))))) * 1.0;
	double tmp;
	if (t_m <= 0.28) {
		tmp = (t_1 * sqrt((z + z))) * 1.0;
	} else if (t_m <= 4.8e+56) {
		tmp = t_2;
	} else if (t_m <= 2e+118) {
		tmp = (t_1 * (z * sqrt(((z + z) / (z * z))))) * 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

t_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    t_2 = (t_1 * sqrt(sqrt(((z + z) * (z + z))))) * 1.0d0
    if (t_m <= 0.28d0) then
        tmp = (t_1 * sqrt((z + z))) * 1.0d0
    else if (t_m <= 4.8d+56) then
        tmp = t_2
    else if (t_m <= 2d+118) then
        tmp = (t_1 * (z * sqrt(((z + z) / (z * z))))) * 1.0d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double t_2 = (t_1 * Math.sqrt(Math.sqrt(((z + z) * (z + z))))) * 1.0;
	double tmp;
	if (t_m <= 0.28) {
		tmp = (t_1 * Math.sqrt((z + z))) * 1.0;
	} else if (t_m <= 4.8e+56) {
		tmp = t_2;
	} else if (t_m <= 2e+118) {
		tmp = (t_1 * (z * Math.sqrt(((z + z) / (z * z))))) * 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = (x * 0.5) - y
	t_2 = (t_1 * math.sqrt(math.sqrt(((z + z) * (z + z))))) * 1.0
	tmp = 0
	if t_m <= 0.28:
		tmp = (t_1 * math.sqrt((z + z))) * 1.0
	elif t_m <= 4.8e+56:
		tmp = t_2
	elif t_m <= 2e+118:
		tmp = (t_1 * (z * math.sqrt(((z + z) / (z * z))))) * 1.0
	else:
		tmp = t_2
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = Float64(Float64(t_1 * sqrt(sqrt(Float64(Float64(z + z) * Float64(z + z))))) * 1.0)
	tmp = 0.0
	if (t_m <= 0.28)
		tmp = Float64(Float64(t_1 * sqrt(Float64(z + z))) * 1.0);
	elseif (t_m <= 4.8e+56)
		tmp = t_2;
	elseif (t_m <= 2e+118)
		tmp = Float64(Float64(t_1 * Float64(z * sqrt(Float64(Float64(z + z) / Float64(z * z))))) * 1.0);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = (x * 0.5) - y;
	t_2 = (t_1 * sqrt(sqrt(((z + z) * (z + z))))) * 1.0;
	tmp = 0.0;
	if (t_m <= 0.28)
		tmp = (t_1 * sqrt((z + z))) * 1.0;
	elseif (t_m <= 4.8e+56)
		tmp = t_2;
	elseif (t_m <= 2e+118)
		tmp = (t_1 * (z * sqrt(((z + z) / (z * z))))) * 1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Sqrt[N[Sqrt[N[(N[(z + z), $MachinePrecision] * N[(z + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[t$95$m, 0.28], N[(N[(t$95$1 * N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$m, 4.8e+56], t$95$2, If[LessEqual[t$95$m, 2e+118], N[(N[(t$95$1 * N[(z * N[Sqrt[N[(N[(z + z), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 0.28000000000000003

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 57.6%

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

   if 0.28000000000000003 < t < 4.80000000000000027e56 or 1.99999999999999993e118 < t

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 46.7%

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

   if 4.80000000000000027e56 < t < 1.99999999999999993e118

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 57.5%

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

   9. Applied rewrites 40.6%

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

Alternative 8: 65.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \left(t\_1 \cdot \sqrt{\sqrt{\left(z + z\right) \cdot \left(z + z\right)}}\right) \cdot 1\\ \mathbf{if}\;t\_m \leq 0.28:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{z + z}\right) \cdot 1\\ \mathbf{elif}\;t\_m \leq 3.25 \cdot 10^{+56}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 3.5 \cdot 10^{+129}:\\ \;\;\;\;\left(z \cdot \left(\sqrt{\frac{2}{z \cdot z} \cdot z} \cdot \left(0.5 \cdot x - y\right)\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y))
        (t_2 (* (* t_1 (sqrt (sqrt (* (+ z z) (+ z z))))) 1.0)))
   (if (<= t_m 0.28)
     (* (* t_1 (sqrt (+ z z))) 1.0)
     (if (<= t_m 3.25e+56)
       t_2
       (if (<= t_m 3.5e+129)
         (* (* z (* (sqrt (* (/ 2.0 (* z z)) z)) (- (* 0.5 x) y))) 1.0)
         t_2)))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double t_2 = (t_1 * sqrt(sqrt(((z + z) * (z + z))))) * 1.0;
	double tmp;
	if (t_m <= 0.28) {
		tmp = (t_1 * sqrt((z + z))) * 1.0;
	} else if (t_m <= 3.25e+56) {
		tmp = t_2;
	} else if (t_m <= 3.5e+129) {
		tmp = (z * (sqrt(((2.0 / (z * z)) * z)) * ((0.5 * x) - y))) * 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

t_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    t_2 = (t_1 * sqrt(sqrt(((z + z) * (z + z))))) * 1.0d0
    if (t_m <= 0.28d0) then
        tmp = (t_1 * sqrt((z + z))) * 1.0d0
    else if (t_m <= 3.25d+56) then
        tmp = t_2
    else if (t_m <= 3.5d+129) then
        tmp = (z * (sqrt(((2.0d0 / (z * z)) * z)) * ((0.5d0 * x) - y))) * 1.0d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double t_2 = (t_1 * Math.sqrt(Math.sqrt(((z + z) * (z + z))))) * 1.0;
	double tmp;
	if (t_m <= 0.28) {
		tmp = (t_1 * Math.sqrt((z + z))) * 1.0;
	} else if (t_m <= 3.25e+56) {
		tmp = t_2;
	} else if (t_m <= 3.5e+129) {
		tmp = (z * (Math.sqrt(((2.0 / (z * z)) * z)) * ((0.5 * x) - y))) * 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = (x * 0.5) - y
	t_2 = (t_1 * math.sqrt(math.sqrt(((z + z) * (z + z))))) * 1.0
	tmp = 0
	if t_m <= 0.28:
		tmp = (t_1 * math.sqrt((z + z))) * 1.0
	elif t_m <= 3.25e+56:
		tmp = t_2
	elif t_m <= 3.5e+129:
		tmp = (z * (math.sqrt(((2.0 / (z * z)) * z)) * ((0.5 * x) - y))) * 1.0
	else:
		tmp = t_2
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = Float64(Float64(t_1 * sqrt(sqrt(Float64(Float64(z + z) * Float64(z + z))))) * 1.0)
	tmp = 0.0
	if (t_m <= 0.28)
		tmp = Float64(Float64(t_1 * sqrt(Float64(z + z))) * 1.0);
	elseif (t_m <= 3.25e+56)
		tmp = t_2;
	elseif (t_m <= 3.5e+129)
		tmp = Float64(Float64(z * Float64(sqrt(Float64(Float64(2.0 / Float64(z * z)) * z)) * Float64(Float64(0.5 * x) - y))) * 1.0);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = (x * 0.5) - y;
	t_2 = (t_1 * sqrt(sqrt(((z + z) * (z + z))))) * 1.0;
	tmp = 0.0;
	if (t_m <= 0.28)
		tmp = (t_1 * sqrt((z + z))) * 1.0;
	elseif (t_m <= 3.25e+56)
		tmp = t_2;
	elseif (t_m <= 3.5e+129)
		tmp = (z * (sqrt(((2.0 / (z * z)) * z)) * ((0.5 * x) - y))) * 1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Sqrt[N[Sqrt[N[(N[(z + z), $MachinePrecision] * N[(z + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[t$95$m, 0.28], N[(N[(t$95$1 * N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$m, 3.25e+56], t$95$2, If[LessEqual[t$95$m, 3.5e+129], N[(N[(z * N[(N[Sqrt[N[(N[(2.0 / N[(z * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 0.28000000000000003

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 57.6%

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

   if 0.28000000000000003 < t < 3.25e56 or 3.4999999999999998e129 < t

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 46.7%

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

   if 3.25e56 < t < 3.4999999999999998e129

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 57.0%

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

   9. Applied rewrites 40.8%

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

Alternative 9: 65.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t\_m \leq 1.14 \cdot 10^{-7}:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{z + z}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{\sqrt{\left(z + z\right) \cdot \left(z + z\right)}}\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t_m 1.14e-7)
     (* (* t_1 (sqrt (+ z z))) 1.0)
     (* (* t_1 (sqrt (sqrt (* (+ z z) (+ z z))))) 1.0))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t_m <= 1.14e-7) {
		tmp = (t_1 * sqrt((z + z))) * 1.0;
	} else {
		tmp = (t_1 * sqrt(sqrt(((z + z) * (z + z))))) * 1.0;
	}
	return tmp;
}

Fortran

t_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if (t_m <= 1.14d-7) then
        tmp = (t_1 * sqrt((z + z))) * 1.0d0
    else
        tmp = (t_1 * sqrt(sqrt(((z + z) * (z + z))))) * 1.0d0
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t_m <= 1.14e-7) {
		tmp = (t_1 * Math.sqrt((z + z))) * 1.0;
	} else {
		tmp = (t_1 * Math.sqrt(Math.sqrt(((z + z) * (z + z))))) * 1.0;
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t_m <= 1.14e-7:
		tmp = (t_1 * math.sqrt((z + z))) * 1.0
	else:
		tmp = (t_1 * math.sqrt(math.sqrt(((z + z) * (z + z))))) * 1.0
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t_m <= 1.14e-7)
		tmp = Float64(Float64(t_1 * sqrt(Float64(z + z))) * 1.0);
	else
		tmp = Float64(Float64(t_1 * sqrt(sqrt(Float64(Float64(z + z) * Float64(z + z))))) * 1.0);
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t_m <= 1.14e-7)
		tmp = (t_1 * sqrt((z + z))) * 1.0;
	else
		tmp = (t_1 * sqrt(sqrt(((z + z) * (z + z))))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t$95$m, 1.14e-7], N[(N[(t$95$1 * N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(t$95$1 * N[Sqrt[N[Sqrt[N[(N[(z + z), $MachinePrecision] * N[(z + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.14000000000000002e-7

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 57.6%

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

   if 1.14000000000000002e-7 < t

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 46.7%

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

Alternative 10: 62.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;e^{\frac{t\_m \cdot t\_m}{2}} \leq 1.02:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{z + z}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 \cdot z\right) \cdot \sqrt{\frac{2}{z}}\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (exp (/ (* t_m t_m) 2.0)) 1.02)
     (* (* t_1 (sqrt (+ z z))) 1.0)
     (* (* (* t_1 z) (sqrt (/ 2.0 z))) 1.0))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (exp(((t_m * t_m) / 2.0)) <= 1.02) {
		tmp = (t_1 * sqrt((z + z))) * 1.0;
	} else {
		tmp = ((t_1 * z) * sqrt((2.0 / z))) * 1.0;
	}
	return tmp;
}

Fortran

t_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if (exp(((t_m * t_m) / 2.0d0)) <= 1.02d0) then
        tmp = (t_1 * sqrt((z + z))) * 1.0d0
    else
        tmp = ((t_1 * z) * sqrt((2.0d0 / z))) * 1.0d0
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (Math.exp(((t_m * t_m) / 2.0)) <= 1.02) {
		tmp = (t_1 * Math.sqrt((z + z))) * 1.0;
	} else {
		tmp = ((t_1 * z) * Math.sqrt((2.0 / z))) * 1.0;
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = (x * 0.5) - y
	tmp = 0
	if math.exp(((t_m * t_m) / 2.0)) <= 1.02:
		tmp = (t_1 * math.sqrt((z + z))) * 1.0
	else:
		tmp = ((t_1 * z) * math.sqrt((2.0 / z))) * 1.0
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (exp(Float64(Float64(t_m * t_m) / 2.0)) <= 1.02)
		tmp = Float64(Float64(t_1 * sqrt(Float64(z + z))) * 1.0);
	else
		tmp = Float64(Float64(Float64(t_1 * z) * sqrt(Float64(2.0 / z))) * 1.0);
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (exp(((t_m * t_m) / 2.0)) <= 1.02)
		tmp = (t_1 * sqrt((z + z))) * 1.0;
	else
		tmp = ((t_1 * z) * sqrt((2.0 / z))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[N[Exp[N[(N[(t$95$m * t$95$m), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 1.02], N[(N[(t$95$1 * N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(t$95$1 * z), $MachinePrecision] * N[Sqrt[N[(2.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (/.f64 (*.f64 t t) #s(literal 2 binary64))) < 1.02

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 57.6%

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

   if 1.02 < (exp.f64 (/.f64 (*.f64 t t) #s(literal 2 binary64)))

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 57.0%

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

   9. Applied rewrites 56.0%

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

Alternative 11: 62.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;e^{\frac{t\_m \cdot t\_m}{2}} \leq 1.02:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{z + z}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{\frac{2}{z}} \cdot t\_1\right) \cdot z\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (exp (/ (* t_m t_m) 2.0)) 1.02)
     (* (* t_1 (sqrt (+ z z))) 1.0)
     (* (* (* (sqrt (/ 2.0 z)) t_1) z) 1.0))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (exp(((t_m * t_m) / 2.0)) <= 1.02) {
		tmp = (t_1 * sqrt((z + z))) * 1.0;
	} else {
		tmp = ((sqrt((2.0 / z)) * t_1) * z) * 1.0;
	}
	return tmp;
}

Fortran

t_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if (exp(((t_m * t_m) / 2.0d0)) <= 1.02d0) then
        tmp = (t_1 * sqrt((z + z))) * 1.0d0
    else
        tmp = ((sqrt((2.0d0 / z)) * t_1) * z) * 1.0d0
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (Math.exp(((t_m * t_m) / 2.0)) <= 1.02) {
		tmp = (t_1 * Math.sqrt((z + z))) * 1.0;
	} else {
		tmp = ((Math.sqrt((2.0 / z)) * t_1) * z) * 1.0;
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = (x * 0.5) - y
	tmp = 0
	if math.exp(((t_m * t_m) / 2.0)) <= 1.02:
		tmp = (t_1 * math.sqrt((z + z))) * 1.0
	else:
		tmp = ((math.sqrt((2.0 / z)) * t_1) * z) * 1.0
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (exp(Float64(Float64(t_m * t_m) / 2.0)) <= 1.02)
		tmp = Float64(Float64(t_1 * sqrt(Float64(z + z))) * 1.0);
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(2.0 / z)) * t_1) * z) * 1.0);
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (exp(((t_m * t_m) / 2.0)) <= 1.02)
		tmp = (t_1 * sqrt((z + z))) * 1.0;
	else
		tmp = ((sqrt((2.0 / z)) * t_1) * z) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[N[Exp[N[(N[(t$95$m * t$95$m), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 1.02], N[(N[(t$95$1 * N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(2.0 / z), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * z), $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (/.f64 (*.f64 t t) #s(literal 2 binary64))) < 1.02

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 57.6%

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

   if 1.02 < (exp.f64 (/.f64 (*.f64 t t) #s(literal 2 binary64)))

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 57.0%

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

   9. Applied rewrites 57.0%

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

Alternative 12: 57.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 1.85 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z + z}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(0.5 \cdot \left(x \cdot \sqrt{\frac{2}{z}}\right)\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (if (<= t_m 1.85e+20)
   (* (* (- (* x 0.5) y) (sqrt (+ z z))) 1.0)
   (* (* z (* 0.5 (* x (sqrt (/ 2.0 z))))) 1.0)))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 1.85e+20) {
		tmp = (((x * 0.5) - y) * sqrt((z + z))) * 1.0;
	} else {
		tmp = (z * (0.5 * (x * sqrt((2.0 / z))))) * 1.0;
	}
	return tmp;
}

Fortran

t_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.85d+20) then
        tmp = (((x * 0.5d0) - y) * sqrt((z + z))) * 1.0d0
    else
        tmp = (z * (0.5d0 * (x * sqrt((2.0d0 / z))))) * 1.0d0
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 1.85e+20) {
		tmp = (((x * 0.5) - y) * Math.sqrt((z + z))) * 1.0;
	} else {
		tmp = (z * (0.5 * (x * Math.sqrt((2.0 / z))))) * 1.0;
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	tmp = 0
	if t_m <= 1.85e+20:
		tmp = (((x * 0.5) - y) * math.sqrt((z + z))) * 1.0
	else:
		tmp = (z * (0.5 * (x * math.sqrt((2.0 / z))))) * 1.0
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	tmp = 0.0
	if (t_m <= 1.85e+20)
		tmp = Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z + z))) * 1.0);
	else
		tmp = Float64(Float64(z * Float64(0.5 * Float64(x * sqrt(Float64(2.0 / z))))) * 1.0);
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	tmp = 0.0;
	if (t_m <= 1.85e+20)
		tmp = (((x * 0.5) - y) * sqrt((z + z))) * 1.0;
	else
		tmp = (z * (0.5 * (x * sqrt((2.0 / z))))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := If[LessEqual[t$95$m, 1.85e+20], N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(z * N[(0.5 * N[(x * N[Sqrt[N[(2.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.85e20

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 57.6%

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

   if 1.85e20 < t

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 57.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 30.1%

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

Alternative 13: 57.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z + z}\right) \cdot 1 \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (+ z z))) 1.0))

C

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

Fortran

t_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    code = (((x * 0.5d0) - y) * sqrt((z + z))) * 1.0d0
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	return (((x * 0.5) - y) * Math.sqrt((z + z))) * 1.0;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	return (((x * 0.5) - y) * math.sqrt((z + z))) * 1.0

Julia

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

MATLAB

t_m = abs(t);
function tmp = code(x, y, z, t_m)
	tmp = (((x * 0.5) - y) * sqrt((z + z))) * 1.0;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in t around 0

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

4. Applied rewrites 57.6%

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

6. Applied rewrites 57.6%

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

Alternative 14: 43.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := \left(\left(\sqrt{z + z} \cdot 0.5\right) \cdot x\right) \cdot 1\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{+129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(-1 \cdot y\right) \cdot \sqrt{z \cdot 2}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (* (* (* (sqrt (+ z z)) 0.5) x) 1.0)))
   (if (<= x -3.1e+129)
     t_1
     (if (<= x 1.4e-21) (* (* (* -1.0 y) (sqrt (* z 2.0))) 1.0) t_1))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = ((sqrt((z + z)) * 0.5) * x) * 1.0;
	double tmp;
	if (x <= -3.1e+129) {
		tmp = t_1;
	} else if (x <= 1.4e-21) {
		tmp = ((-1.0 * y) * sqrt((z * 2.0))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

t_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((sqrt((z + z)) * 0.5d0) * x) * 1.0d0
    if (x <= (-3.1d+129)) then
        tmp = t_1
    else if (x <= 1.4d-21) then
        tmp = (((-1.0d0) * y) * sqrt((z * 2.0d0))) * 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = ((Math.sqrt((z + z)) * 0.5) * x) * 1.0;
	double tmp;
	if (x <= -3.1e+129) {
		tmp = t_1;
	} else if (x <= 1.4e-21) {
		tmp = ((-1.0 * y) * Math.sqrt((z * 2.0))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = ((math.sqrt((z + z)) * 0.5) * x) * 1.0
	tmp = 0
	if x <= -3.1e+129:
		tmp = t_1
	elif x <= 1.4e-21:
		tmp = ((-1.0 * y) * math.sqrt((z * 2.0))) * 1.0
	else:
		tmp = t_1
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = Float64(Float64(Float64(sqrt(Float64(z + z)) * 0.5) * x) * 1.0)
	tmp = 0.0
	if (x <= -3.1e+129)
		tmp = t_1;
	elseif (x <= 1.4e-21)
		tmp = Float64(Float64(Float64(-1.0 * y) * sqrt(Float64(z * 2.0))) * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = ((sqrt((z + z)) * 0.5) * x) * 1.0;
	tmp = 0.0;
	if (x <= -3.1e+129)
		tmp = t_1;
	elseif (x <= 1.4e-21)
		tmp = ((-1.0 * y) * sqrt((z * 2.0))) * 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[(N[(N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[x, -3.1e+129], t$95$1, If[LessEqual[x, 1.4e-21], N[(N[(N[(-1.0 * y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1e129 or 1.40000000000000002e-21 < x

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 30.1%

      \[\leadsto \color{blue}{\left(0.5 \cdot \left(x \cdot \sqrt{2 \cdot z}\right)\right)} \cdot 1 \]

   9. Applied rewrites 30.1%

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

   if -3.1e129 < x < 1.40000000000000002e-21

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 30.0%

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

Alternative 15: 30.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \left(\left(\sqrt{z + z} \cdot 0.5\right) \cdot x\right) \cdot 1 \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m) :precision binary64 (* (* (* (sqrt (+ z z)) 0.5) x) 1.0))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	return ((sqrt((z + z)) * 0.5) * x) * 1.0;
}

Fortran

t_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    code = ((sqrt((z + z)) * 0.5d0) * x) * 1.0d0
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	return ((Math.sqrt((z + z)) * 0.5) * x) * 1.0;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	return ((math.sqrt((z + z)) * 0.5) * x) * 1.0

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	return Float64(Float64(Float64(sqrt(Float64(z + z)) * 0.5) * x) * 1.0)
end

MATLAB

t_m = abs(t);
function tmp = code(x, y, z, t_m)
	tmp = ((sqrt((z + z)) * 0.5) * x) * 1.0;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[(N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in t around 0

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

4. Applied rewrites 57.6%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 30.1%

   \[\leadsto \color{blue}{\left(0.5 \cdot \left(x \cdot \sqrt{2 \cdot z}\right)\right)} \cdot 1 \]

9. Applied rewrites 30.1%

   \[\leadsto \left(\left(\sqrt{z + z} \cdot 0.5\right) \cdot \color{blue}{x}\right) \cdot 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025153 
(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))))