System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2

Percentage Accurate: 61.1% → 92.6%

Time: 13.7s

Alternatives: 11

Speedup: 8.7×

• could not determine a ground truth

  1. x = -1.602183383368253e-237
  2. y = 1.8524875282262932e+97
  3. z = 4.323197761916066e+289
  4. t = 7.303707345373385e+119

Specification

Math

\[\begin{array}{l} \\ x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))

C

double code(double x, double y, double z, double t) {
	return x - (log(((1.0 - y) + (y * exp(z)))) / t);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (log(((1.0d0 - y) + (y * exp(z)))) / t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (Math.log(((1.0 - y) + (y * Math.exp(z)))) / t);
}

Python

def code(x, y, z, t):
	return x - (math.log(((1.0 - y) + (y * math.exp(z)))) / t)

Julia

function code(x, y, z, t)
	return Float64(x - Float64(log(Float64(Float64(1.0 - y) + Float64(y * exp(z)))) / t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (log(((1.0 - y) + (y * exp(z)))) / t);
end

Wolfram

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

Initial Program: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))

C

double code(double x, double y, double z, double t) {
	return x - (log(((1.0 - y) + (y * exp(z)))) / t);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (log(((1.0d0 - y) + (y * exp(z)))) / t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (Math.log(((1.0 - y) + (y * Math.exp(z)))) / t);
}

Python

def code(x, y, z, t):
	return x - (math.log(((1.0 - y) + (y * math.exp(z)))) / t)

Julia

function code(x, y, z, t)
	return Float64(x - Float64(log(Float64(Float64(1.0 - y) + Float64(y * exp(z)))) / t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (log(((1.0 - y) + (y * exp(z)))) / t);
end

Wolfram

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

Alternative 1: 92.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+23}:\\ \;\;\;\;\frac{t \cdot x - \mathsf{log1p}\left(\mathsf{expm1}\left(z\right) \cdot y\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.45e+23)
   (/ (- (* t x) (log1p (* (expm1 z) y))) t)
   (- x (* (/ (expm1 z) t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.45e+23) {
		tmp = ((t * x) - log1p((expm1(z) * y))) / t;
	} else {
		tmp = x - ((expm1(z) / t) * y);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.45e+23) {
		tmp = ((t * x) - Math.log1p((Math.expm1(z) * y))) / t;
	} else {
		tmp = x - ((Math.expm1(z) / t) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1.45e+23:
		tmp = ((t * x) - math.log1p((math.expm1(z) * y))) / t
	else:
		tmp = x - ((math.expm1(z) / t) * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.45e+23)
		tmp = Float64(Float64(Float64(t * x) - log1p(Float64(expm1(z) * y))) / t);
	else
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.45e+23], N[(N[(N[(t * x), $MachinePrecision] - N[Log[1 + N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.45000000000000006e23

   1. Initial program 43.0%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\frac{t \cdot x - \mathsf{log1p}\left(\mathsf{expm1}\left(z\right) \cdot y\right)}{t}} \]

   if -1.45000000000000006e23 < y

   1. Initial program 63.2%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 95.3

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 95.3%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 92.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(\left(1 - y\right) + y \cdot e^{z}\right) \leq 0:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{expm1}\left(z\right) \cdot y\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (log (+ (- 1.0 y) (* y (exp z)))) 0.0)
   (- x (* (/ (expm1 z) t) y))
   (- x (/ (log (* (expm1 z) y)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (log(((1.0 - y) + (y * exp(z)))) <= 0.0) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (log((expm1(z) * y)) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (Math.log(((1.0 - y) + (y * Math.exp(z)))) <= 0.0) {
		tmp = x - ((Math.expm1(z) / t) * y);
	} else {
		tmp = x - (Math.log((Math.expm1(z) * y)) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if math.log(((1.0 - y) + (y * math.exp(z)))) <= 0.0:
		tmp = x - ((math.expm1(z) / t) * y)
	else:
		tmp = x - (math.log((math.expm1(z) * y)) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (log(Float64(Float64(1.0 - y) + Float64(y * exp(z)))) <= 0.0)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(log(Float64(expm1(z) * y)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Log[N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))) < 0.0

   1. Initial program 54.2%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 93.3

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 93.3%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]

   if 0.0 < (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))))

   1. Initial program 95.3%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. distribute-lft-neg-out N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-expm1.f64 96.3

          \[\leadsto x - \frac{\log \left(\color{blue}{\mathsf{expm1}\left(z\right)} \cdot y\right)}{t} \]

   5. Applied rewrites 96.3%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{expm1}\left(z\right) \cdot y\right)}}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(\left(1 - y\right) + y \cdot e^{z}\right) \leq 85:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (log (+ (- 1.0 y) (* y (exp z)))) 85.0)
   (- x (* (/ (expm1 z) t) y))
   (- x (/ (log 1.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (log(((1.0 - y) + (y * exp(z)))) <= 85.0) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (log(1.0) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (Math.log(((1.0 - y) + (y * Math.exp(z)))) <= 85.0) {
		tmp = x - ((Math.expm1(z) / t) * y);
	} else {
		tmp = x - (Math.log(1.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if math.log(((1.0 - y) + (y * math.exp(z)))) <= 85.0:
		tmp = x - ((math.expm1(z) / t) * y)
	else:
		tmp = x - (math.log(1.0) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (log(Float64(Float64(1.0 - y) + Float64(y * exp(z)))) <= 85.0)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(log(1.0) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Log[N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 85.0], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))) < 85

   1. Initial program 55.0%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 92.7

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 92.7%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]

   if 85 < (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))))

   1. Initial program 94.5%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]
   4. Step-by-step derivation

   5. Applied rewrites 59.2%

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+35}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -8e+35)
   (- x (/ (log (fma z y 1.0)) t))
   (- x (* (/ (expm1 z) t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -8e+35) {
		tmp = x - (log(fma(z, y, 1.0)) / t);
	} else {
		tmp = x - ((expm1(z) / t) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -8e+35)
		tmp = Float64(x - Float64(log(fma(z, y, 1.0)) / t));
	else
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -8e+35], N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.9999999999999997e35

   1. Initial program 44.2%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 71.2

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

   5. Applied rewrites 71.2%

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

   if -7.9999999999999997e35 < y

   1. Initial program 62.5%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 94.8

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 94.8%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 81.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+21}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.16666666666666666, z, 0.5\right)}{t}, z, \frac{1}{t}\right) \cdot z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -5.4e+21)
   (- x (/ (log 1.0) t))
   (- x (* (* (fma (/ (fma 0.16666666666666666 z 0.5) t) z (/ 1.0 t)) z) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.4e+21) {
		tmp = x - (log(1.0) / t);
	} else {
		tmp = x - ((fma((fma(0.16666666666666666, z, 0.5) / t), z, (1.0 / t)) * z) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -5.4e+21)
		tmp = Float64(x - Float64(log(1.0) / t));
	else
		tmp = Float64(x - Float64(Float64(fma(Float64(fma(0.16666666666666666, z, 0.5) / t), z, Float64(1.0 / t)) * z) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -5.4e+21], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(N[(N[(0.16666666666666666 * z + 0.5), $MachinePrecision] / t), $MachinePrecision] * z + N[(1.0 / t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4e21

   1. Initial program 77.1%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]
   4. Step-by-step derivation

   5. Applied rewrites 65.5%

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]

   if -5.4e21 < z

   1. Initial program 52.2%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 89.5

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 89.5%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 90.0%

      \[\leadsto x - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.16666666666666666, z, 0.5\right)}{t}, z, \frac{1}{t}\right) \cdot z\right) \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 79.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{z} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.16666666666666666, z, 0.5\right)}{t}, z, \frac{1}{t}\right) \cdot z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -5.4e+21)
   (* (/ x z) z)
   (- x (* (* (fma (/ (fma 0.16666666666666666 z 0.5) t) z (/ 1.0 t)) z) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.4e+21) {
		tmp = (x / z) * z;
	} else {
		tmp = x - ((fma((fma(0.16666666666666666, z, 0.5) / t), z, (1.0 / t)) * z) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -5.4e+21)
		tmp = Float64(Float64(x / z) * z);
	else
		tmp = Float64(x - Float64(Float64(fma(Float64(fma(0.16666666666666666, z, 0.5) / t), z, Float64(1.0 / t)) * z) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -5.4e+21], N[(N[(x / z), $MachinePrecision] * z), $MachinePrecision], N[(x - N[(N[(N[(N[(N[(0.16666666666666666 * z + 0.5), $MachinePrecision] / t), $MachinePrecision] * z + N[(1.0 / t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4e21

   1. Initial program 77.1%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 24.6%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 21.8%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{x}{z} \cdot z \]
   10. Step-by-step derivation

   11. Applied rewrites 61.0%

       \[\leadsto \frac{x}{z} \cdot z \]

   if -5.4e21 < z

   1. Initial program 52.2%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 89.5

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 89.5%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 90.0%

      \[\leadsto x - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.16666666666666666, z, 0.5\right)}{t}, z, \frac{1}{t}\right) \cdot z\right) \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 79.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{z} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right), z, 1\right)}{t} \cdot z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -5.4e+21)
   (* (/ x z) z)
   (- x (* (* (/ (fma (fma 0.16666666666666666 z 0.5) z 1.0) t) z) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.4e+21) {
		tmp = (x / z) * z;
	} else {
		tmp = x - (((fma(fma(0.16666666666666666, z, 0.5), z, 1.0) / t) * z) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -5.4e+21)
		tmp = Float64(Float64(x / z) * z);
	else
		tmp = Float64(x - Float64(Float64(Float64(fma(fma(0.16666666666666666, z, 0.5), z, 1.0) / t) * z) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -5.4e+21], N[(N[(x / z), $MachinePrecision] * z), $MachinePrecision], N[(x - N[(N[(N[(N[(N[(0.16666666666666666 * z + 0.5), $MachinePrecision] * z + 1.0), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4e21

   1. Initial program 77.1%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 24.6%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 21.8%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{x}{z} \cdot z \]
   10. Step-by-step derivation

   11. Applied rewrites 61.0%

       \[\leadsto \frac{x}{z} \cdot z \]

   if -5.4e21 < z

   1. Initial program 52.2%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 89.5

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 89.5%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 89.8%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 90.0%

       \[\leadsto x - \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right), z, 1\right)}{t} \cdot z\right) \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 79.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.9:\\ \;\;\;\;\frac{x}{z} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{\mathsf{fma}\left(0.5, z, 1\right)}{t} \cdot z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -0.9) (* (/ x z) z) (- x (* (* (/ (fma 0.5 z 1.0) t) z) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.9) {
		tmp = (x / z) * z;
	} else {
		tmp = x - (((fma(0.5, z, 1.0) / t) * z) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -0.9)
		tmp = Float64(Float64(x / z) * z);
	else
		tmp = Float64(x - Float64(Float64(Float64(fma(0.5, z, 1.0) / t) * z) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -0.9], N[(N[(x / z), $MachinePrecision] * z), $MachinePrecision], N[(x - N[(N[(N[(N[(0.5 * z + 1.0), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.900000000000000022

   1. Initial program 79.2%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 27.7%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 25.1%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{x}{z} \cdot z \]
   10. Step-by-step derivation

   11. Applied rewrites 60.8%

       \[\leadsto \frac{x}{z} \cdot z \]

   if -0.900000000000000022 < z

   1. Initial program 50.3%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 90.6

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 90.6%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 91.1%

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

Alternative 9: 79.2% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+54}:\\ \;\;\;\;\frac{x}{z} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -4.4e+54) (* (/ x z) z) (- x (* (/ z t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.4e+54) {
		tmp = (x / z) * z;
	} else {
		tmp = x - ((z / t) * y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-4.4d+54)) then
        tmp = (x / z) * z
    else
        tmp = x - ((z / t) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.4e+54) {
		tmp = (x / z) * z;
	} else {
		tmp = x - ((z / t) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -4.4e+54:
		tmp = (x / z) * z
	else:
		tmp = x - ((z / t) * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -4.4e+54)
		tmp = Float64(Float64(x / z) * z);
	else
		tmp = Float64(x - Float64(Float64(z / t) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -4.4e+54)
		tmp = (x / z) * z;
	else
		tmp = x - ((z / t) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -4.4e+54], N[(N[(x / z), $MachinePrecision] * z), $MachinePrecision], N[(x - N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.3999999999999998e54

   1. Initial program 74.5%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 19.2%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 16.1%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{x}{z} \cdot z \]
   10. Step-by-step derivation

   11. Applied rewrites 58.1%

       \[\leadsto \frac{x}{z} \cdot z \]

   if -4.3999999999999998e54 < z

   1. Initial program 53.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 89.4

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 89.4%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]

   6. Taylor expanded in z around 0

      \[\leadsto x - \frac{z}{t} \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 89.5%

      \[\leadsto x - \frac{z}{t} \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 76.1% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{z} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{y}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.22e+55) (* (/ x z) z) (fma (- z) (/ y t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.22e+55) {
		tmp = (x / z) * z;
	} else {
		tmp = fma(-z, (y / t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.22e+55)
		tmp = Float64(Float64(x / z) * z);
	else
		tmp = fma(Float64(-z), Float64(y / t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.22e+55], N[(N[(x / z), $MachinePrecision] * z), $MachinePrecision], N[((-z) * N[(y / t), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.22e55

   1. Initial program 74.5%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 19.2%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 16.1%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{x}{z} \cdot z \]
   10. Step-by-step derivation

   11. Applied rewrites 58.1%

       \[\leadsto \frac{x}{z} \cdot z \]

   if -1.22e55 < z

   1. Initial program 53.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{y \cdot \frac{e^{z} - 1}{t}} \]

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{e^{z} - 1}{t}} \cdot y \]

      7. lower-expm1.f64 89.4

         \[\leadsto x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \cdot y \]

   5. Applied rewrites 89.4%

      \[\leadsto x - \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{t} \cdot y} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y \cdot z}{t}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot \frac{y}{t} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(z\right), \frac{y}{t}, x\right)} \]

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 85.4

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

   8. Applied rewrites 85.4%

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

Alternative 11: 53.0% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{z} \cdot z \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (/ x z) z))

C

double code(double x, double y, double z, double t) {
	return (x / z) * z;
}

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 / z) * z
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x / z) * z;
}

Python

def code(x, y, z, t):
	return (x / z) * z

Julia

function code(x, y, z, t)
	return Float64(Float64(x / z) * z)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x / z) * z;
end

Wolfram

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

Derivation

1. Initial program 58.7%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 62.2%

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

6. Taylor expanded in z around -inf

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

8. Applied rewrites 35.5%

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

9. Taylor expanded in x around inf

   \[\leadsto \frac{x}{z} \cdot z \]
10. Step-by-step derivation

11. Applied rewrites 50.9%

    \[\leadsto \frac{x}{z} \cdot z \]
12. Add Preprocessing

Developer Target 1: 74.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-0.5}{y \cdot t}\\ \mathbf{if}\;z < -2.8874623088207947 \cdot 10^{+119}:\\ \;\;\;\;\left(x - \frac{t\_1}{z \cdot z}\right) - t\_1 \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- 0.5) (* y t))))
   (if (< z -2.8874623088207947e+119)
     (- (- x (/ t_1 (* z z))) (* t_1 (/ (/ 2.0 z) (* z z))))
     (- x (/ (log (+ 1.0 (* z y))) t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -0.5 / (y * t);
	double tmp;
	if (z < -2.8874623088207947e+119) {
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	} else {
		tmp = x - (log((1.0 + (z * y))) / t);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -0.5d0 / (y * t)
    if (z < (-2.8874623088207947d+119)) then
        tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0d0 / z) / (z * z)))
    else
        tmp = x - (log((1.0d0 + (z * y))) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -0.5 / (y * t);
	double tmp;
	if (z < -2.8874623088207947e+119) {
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	} else {
		tmp = x - (Math.log((1.0 + (z * y))) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -0.5 / (y * t)
	tmp = 0
	if z < -2.8874623088207947e+119:
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)))
	else:
		tmp = x - (math.log((1.0 + (z * y))) / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-0.5) / Float64(y * t))
	tmp = 0.0
	if (z < -2.8874623088207947e+119)
		tmp = Float64(Float64(x - Float64(t_1 / Float64(z * z))) - Float64(t_1 * Float64(Float64(2.0 / z) / Float64(z * z))));
	else
		tmp = Float64(x - Float64(log(Float64(1.0 + Float64(z * y))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -0.5 / (y * t);
	tmp = 0.0;
	if (z < -2.8874623088207947e+119)
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	else
		tmp = x - (log((1.0 + (z * y))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-0.5) / N[(y * t), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -2.8874623088207947e+119], N[(N[(x - N[(t$95$1 / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(N[(2.0 / z), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(1.0 + N[(z * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2025008 
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -288746230882079470000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (- x (/ (/ (- 1/2) (* y t)) (* z z))) (* (/ (- 1/2) (* y t)) (/ (/ 2 z) (* z z)))) (- x (/ (log (+ 1 (* z y))) t))))

  (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))