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

Percentage Accurate: 61.8% → 94.5%

Time: 7.8s

Alternatives: 13

Speedup: 226.0×

• could not determine a ground truth

  1. x = -4.97188183423245e-73
  2. y = 5.843066835601204e+247
  3. z = 6.858758133083187e+183
  4. t = -1.543586294876956e+224

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: 61.8% 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: 94.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}{t}\\ \mathbf{if}\;y \leq -0.045:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+114}:\\ \;\;\;\;x - \mathsf{fma}\left(-0.5, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{\mathsf{expm1}\left(z\right)}{t}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (log (fma (expm1 z) y 1.0)) t))))
   (if (<= y -0.045)
     t_1
     (if (<= y 4.1e+114)
       (- x (* (fma -0.5 (/ (* (pow (expm1 z) 2.0) y) t) (/ (expm1 z) t)) y))
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (log(fma(expm1(z), y, 1.0)) / t);
	double tmp;
	if (y <= -0.045) {
		tmp = t_1;
	} else if (y <= 4.1e+114) {
		tmp = x - (fma(-0.5, ((pow(expm1(z), 2.0) * y) / t), (expm1(z) / t)) * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(log(fma(expm1(z), y, 1.0)) / t))
	tmp = 0.0
	if (y <= -0.045)
		tmp = t_1;
	elseif (y <= 4.1e+114)
		tmp = Float64(x - Float64(fma(-0.5, Float64(Float64((expm1(z) ^ 2.0) * y) / t), Float64(expm1(z) / t)) * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[Log[N[(N[(Exp[z] - 1), $MachinePrecision] * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.045], t$95$1, If[LessEqual[y, 4.1e+114], N[(x - N[(N[(-0.5 * N[(N[(N[Power[N[(Exp[z] - 1), $MachinePrecision], 2.0], $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision] + N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.044999999999999998 or 4.1000000000000001e114 < y

   1. Initial program 36.5%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-expm1.f64 86.2

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}{t} \]

   4. Applied rewrites 86.2%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}}{t} \]

   if -0.044999999999999998 < y < 4.1000000000000001e114

   1. Initial program 74.3%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-expm1.f64 N/A

         \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z}}{t} - \frac{1}{t}\right) \cdot y \]

      10. sub-div N/A

          \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z} - 1}{t}\right) \cdot y \]

      11. lower-/.f64 N/A

          \[\leadsto x - \mathsf{fma}\left(\frac{-1}{2}, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{e^{z} - 1}{t}\right) \cdot y \]

      12. lower-expm1.f64 98.6

          \[\leadsto x - \mathsf{fma}\left(-0.5, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{\mathsf{expm1}\left(z\right)}{t}\right) \cdot y \]

   4. Applied rewrites 98.6%

      \[\leadsto x - \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(\mathsf{expm1}\left(z\right)\right)}^{2} \cdot y}{t}, \frac{\mathsf{expm1}\left(z\right)}{t}\right) \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 94.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(\left(1 - y\right) + y \cdot e^{z}\right)\\ t_2 := \mathsf{expm1}\left(z\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-\left(\frac{\mathsf{log1p}\left(z \cdot y\right)}{t \cdot x} - 1\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;x - \frac{t\_2}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log t\_2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (log (+ (- 1.0 y) (* y (exp z))))) (t_2 (* (expm1 z) y)))
   (if (<= t_1 (- INFINITY))
     (- (* (- (/ (log1p (* z y)) (* t x)) 1.0) x))
     (if (<= t_1 0.0) (- x (/ t_2 t)) (- x (/ (log t_2) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(((1.0 - y) + (y * exp(z))));
	double t_2 = expm1(z) * y;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -(((log1p((z * y)) / (t * x)) - 1.0) * x);
	} else if (t_1 <= 0.0) {
		tmp = x - (t_2 / t);
	} else {
		tmp = x - (log(t_2) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(((1.0 - y) + (y * Math.exp(z))));
	double t_2 = Math.expm1(z) * y;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -(((Math.log1p((z * y)) / (t * x)) - 1.0) * x);
	} else if (t_1 <= 0.0) {
		tmp = x - (t_2 / t);
	} else {
		tmp = x - (Math.log(t_2) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(((1.0 - y) + (y * math.exp(z))))
	t_2 = math.expm1(z) * y
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -(((math.log1p((z * y)) / (t * x)) - 1.0) * x)
	elif t_1 <= 0.0:
		tmp = x - (t_2 / t)
	else:
		tmp = x - (math.log(t_2) / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = log(Float64(Float64(1.0 - y) + Float64(y * exp(z))))
	t_2 = Float64(expm1(z) * y)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-Float64(Float64(Float64(log1p(Float64(z * y)) / Float64(t * x)) - 1.0) * x));
	elseif (t_1 <= 0.0)
		tmp = Float64(x - Float64(t_2 / t));
	else
		tmp = Float64(x - Float64(log(t_2) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Log[N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-N[(N[(N[(N[Log[1 + N[(z * y), $MachinePrecision]], $MachinePrecision] / N[(t * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]), If[LessEqual[t$95$1, 0.0], N[(x - N[(t$95$2 / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[t$95$2], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))) < -inf.0

   1. Initial program 2.1%

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

   2. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 59.5%

      \[\leadsto \color{blue}{-\left(\frac{\mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t \cdot x} - 1\right) \cdot x} \]

   5. Taylor expanded in z around 0

      \[\leadsto -\left(\frac{\mathsf{log1p}\left(y \cdot z\right)}{t \cdot x} - 1\right) \cdot x \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -\left(\frac{\mathsf{log1p}\left(z \cdot y\right)}{t \cdot x} - 1\right) \cdot x \]

      2. lift-*.f64 89.1

         \[\leadsto -\left(\frac{\mathsf{log1p}\left(z \cdot y\right)}{t \cdot x} - 1\right) \cdot x \]

   7. Applied rewrites 89.1%

      \[\leadsto -\left(\frac{\mathsf{log1p}\left(z \cdot y\right)}{t \cdot x} - 1\right) \cdot x \]

   if -inf.0 < (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))) < 0.0

   1. Initial program 80.9%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 97.8

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

   4. Applied rewrites 97.8%

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

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

   1. Initial program 93.8%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 92.8

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

   4. Applied rewrites 92.8%

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

Alternative 3: 93.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}{t}\\ \mathbf{if}\;y \leq -0.00035:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+114}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (log (fma (expm1 z) y 1.0)) t))))
   (if (<= y -0.00035)
     t_1
     (if (<= y 4.1e+114) (- x (/ (* (expm1 z) y) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (log(fma(expm1(z), y, 1.0)) / t);
	double tmp;
	if (y <= -0.00035) {
		tmp = t_1;
	} else if (y <= 4.1e+114) {
		tmp = x - ((expm1(z) * y) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(log(fma(expm1(z), y, 1.0)) / t))
	tmp = 0.0
	if (y <= -0.00035)
		tmp = t_1;
	elseif (y <= 4.1e+114)
		tmp = Float64(x - Float64(Float64(expm1(z) * y) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[Log[N[(N[(Exp[z] - 1), $MachinePrecision] * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.00035], t$95$1, If[LessEqual[y, 4.1e+114], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.49999999999999996e-4 or 4.1000000000000001e114 < y

   1. Initial program 36.8%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-expm1.f64 86.2

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}{t} \]

   4. Applied rewrites 86.2%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}}{t} \]

   if -3.49999999999999996e-4 < y < 4.1000000000000001e114

   1. Initial program 74.2%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 97.0

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

   4. Applied rewrites 97.0%

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

Alternative 4: 87.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+170}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+114}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, z, 0.16666666666666666\right), z, 0.5\right), z, 1\right) \cdot z, y, 1\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -7.2e+170)
   (- x (/ (log (fma z y 1.0)) t))
   (if (<= y 4.1e+114)
     (- x (/ (* (expm1 z) y) t))
     (-
      x
      (/
       (log
        (fma
         (*
          (fma
           (fma (fma 0.041666666666666664 z 0.16666666666666666) z 0.5)
           z
           1.0)
          z)
         y
         1.0))
       t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7.2e+170) {
		tmp = x - (log(fma(z, y, 1.0)) / t);
	} else if (y <= 4.1e+114) {
		tmp = x - ((expm1(z) * y) / t);
	} else {
		tmp = x - (log(fma((fma(fma(fma(0.041666666666666664, z, 0.16666666666666666), z, 0.5), z, 1.0) * z), y, 1.0)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -7.2e+170)
		tmp = Float64(x - Float64(log(fma(z, y, 1.0)) / t));
	elseif (y <= 4.1e+114)
		tmp = Float64(x - Float64(Float64(expm1(z) * y) / t));
	else
		tmp = Float64(x - Float64(log(fma(Float64(fma(fma(fma(0.041666666666666664, z, 0.16666666666666666), z, 0.5), z, 1.0) * z), y, 1.0)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -7.2e+170], N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.1e+114], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(N[(N[(N[(N[(0.041666666666666664 * z + 0.16666666666666666), $MachinePrecision] * z + 0.5), $MachinePrecision] * z + 1.0), $MachinePrecision] * z), $MachinePrecision] * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.1999999999999999e170

   1. Initial program 49.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 54.0

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

   4. Applied rewrites 54.0%

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

   if -7.1999999999999999e170 < y < 4.1000000000000001e114

   1. Initial program 69.0%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 92.3

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

   4. Applied rewrites 92.3%

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

   if 4.1000000000000001e114 < y

   1. Initial program 5.6%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-expm1.f64 81.5

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}{t} \]

   4. Applied rewrites 81.5%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}}{t} \]

   5. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(z \cdot \left(1 + z \cdot \left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right)\right)\right), y, 1\right)\right)}{t} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\left(1 + z \cdot \left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right)\right)\right) \cdot z, y, 1\right)\right)}{t} \]

      2. lower-*.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\left(1 + z \cdot \left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right)\right)\right) \cdot z, y, 1\right)\right)}{t} \]

      3. +-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\left(z \cdot \left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right)\right) + 1\right) \cdot z, y, 1\right)\right)}{t} \]

      4. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\left(\left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right)\right) \cdot z + 1\right) \cdot z, y, 1\right)\right)}{t} \]

      5. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} + z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right), z, 1\right) \cdot z, y, 1\right)\right)}{t} \]

      6. +-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot z\right) + \frac{1}{2}, z, 1\right) \cdot z, y, 1\right)\right)}{t} \]

      7. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{1}{6} + \frac{1}{24} \cdot z\right) \cdot z + \frac{1}{2}, z, 1\right) \cdot z, y, 1\right)\right)}{t} \]

      8. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} + \frac{1}{24} \cdot z, z, \frac{1}{2}\right), z, 1\right) \cdot z, y, 1\right)\right)}{t} \]

      9. +-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} \cdot z + \frac{1}{6}, z, \frac{1}{2}\right), z, 1\right) \cdot z, y, 1\right)\right)}{t} \]

      10. lower-fma.f64 82.1

          \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, z, 0.16666666666666666\right), z, 0.5\right), z, 1\right) \cdot z, y, 1\right)\right)}{t} \]

   7. Applied rewrites 82.1%

      \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, z, 0.16666666666666666\right), z, 0.5\right), z, 1\right) \cdot z, y, 1\right)\right)}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 87.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+170}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+114}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, 0.5 \cdot y\right), z, y\right), z, 1\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -7.2e+170)
   (- x (/ (log (fma z y 1.0)) t))
   (if (<= y 4.1e+114)
     (- x (/ (* (expm1 z) y) t))
     (-
      x
      (/
       (log (fma (fma (fma 0.16666666666666666 (* z y) (* 0.5 y)) z y) z 1.0))
       t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7.2e+170) {
		tmp = x - (log(fma(z, y, 1.0)) / t);
	} else if (y <= 4.1e+114) {
		tmp = x - ((expm1(z) * y) / t);
	} else {
		tmp = x - (log(fma(fma(fma(0.16666666666666666, (z * y), (0.5 * y)), z, y), z, 1.0)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -7.2e+170)
		tmp = Float64(x - Float64(log(fma(z, y, 1.0)) / t));
	elseif (y <= 4.1e+114)
		tmp = Float64(x - Float64(Float64(expm1(z) * y) / t));
	else
		tmp = Float64(x - Float64(log(fma(fma(fma(0.16666666666666666, Float64(z * y), Float64(0.5 * y)), z, y), z, 1.0)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -7.2e+170], N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.1e+114], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(N[(N[(0.16666666666666666 * N[(z * y), $MachinePrecision] + N[(0.5 * y), $MachinePrecision]), $MachinePrecision] * z + y), $MachinePrecision] * z + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.1999999999999999e170

   1. Initial program 49.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 54.0

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

   4. Applied rewrites 54.0%

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

   if -7.1999999999999999e170 < y < 4.1000000000000001e114

   1. Initial program 69.0%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 92.3

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

   4. Applied rewrites 92.3%

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

   if 4.1000000000000001e114 < y

   1. Initial program 5.6%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto x - \frac{\log \left(z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right) + \color{blue}{1}\right)}{t} \]

      2. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right) \cdot z + 1\right)}{t} \]

      3. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right), \color{blue}{z}, 1\right)\right)}{t} \]

      4. +-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right) + y, z, 1\right)\right)}{t} \]

      5. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right) \cdot z + y, z, 1\right)\right)}{t} \]

      6. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y, z, y\right), z, 1\right)\right)}{t} \]

      7. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, y \cdot z, \frac{1}{2} \cdot y\right), z, y\right), z, 1\right)\right)}{t} \]

      8. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, z \cdot y, \frac{1}{2} \cdot y\right), z, y\right), z, 1\right)\right)}{t} \]

      9. lower-*.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, z \cdot y, \frac{1}{2} \cdot y\right), z, y\right), z, 1\right)\right)}{t} \]

      10. lower-*.f64 82.0

          \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, 0.5 \cdot y\right), z, y\right), z, 1\right)\right)}{t} \]

   4. Applied rewrites 82.0%

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

Alternative 6: 87.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+170}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+114}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, 0.5, y\right), z, 1\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -7.2e+170)
   (- x (/ (log (fma z y 1.0)) t))
   (if (<= y 4.1e+114)
     (- x (/ (* (expm1 z) y) t))
     (- x (/ (log (fma (fma (* z y) 0.5 y) z 1.0)) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7.2e+170) {
		tmp = x - (log(fma(z, y, 1.0)) / t);
	} else if (y <= 4.1e+114) {
		tmp = x - ((expm1(z) * y) / t);
	} else {
		tmp = x - (log(fma(fma((z * y), 0.5, y), z, 1.0)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -7.2e+170)
		tmp = Float64(x - Float64(log(fma(z, y, 1.0)) / t));
	elseif (y <= 4.1e+114)
		tmp = Float64(x - Float64(Float64(expm1(z) * y) / t));
	else
		tmp = Float64(x - Float64(log(fma(fma(Float64(z * y), 0.5, y), z, 1.0)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -7.2e+170], N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.1e+114], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(N[(N[(z * y), $MachinePrecision] * 0.5 + y), $MachinePrecision] * z + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.1999999999999999e170

   1. Initial program 49.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 54.0

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

   4. Applied rewrites 54.0%

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

   if -7.1999999999999999e170 < y < 4.1000000000000001e114

   1. Initial program 69.0%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 92.3

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

   4. Applied rewrites 92.3%

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

   if 4.1000000000000001e114 < y

   1. Initial program 5.6%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 81.9

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

   4. Applied rewrites 81.9%

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

Alternative 7: 87.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \mathbf{if}\;y \leq -7.2 \cdot 10^{+170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+114}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (log (fma z y 1.0)) t))))
   (if (<= y -7.2e+170)
     t_1
     (if (<= y 4.1e+114) (- x (/ (* (expm1 z) y) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (log(fma(z, y, 1.0)) / t);
	double tmp;
	if (y <= -7.2e+170) {
		tmp = t_1;
	} else if (y <= 4.1e+114) {
		tmp = x - ((expm1(z) * y) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(log(fma(z, y, 1.0)) / t))
	tmp = 0.0
	if (y <= -7.2e+170)
		tmp = t_1;
	elseif (y <= 4.1e+114)
		tmp = Float64(x - Float64(Float64(expm1(z) * y) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.2e+170], t$95$1, If[LessEqual[y, 4.1e+114], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.1999999999999999e170 or 4.1000000000000001e114 < y

   1. Initial program 30.0%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 66.4

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

   4. Applied rewrites 66.4%

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

   if -7.1999999999999999e170 < y < 4.1000000000000001e114

   1. Initial program 69.0%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 92.3

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

   4. Applied rewrites 92.3%

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

Alternative 8: 86.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\log \left(z \cdot y\right)}{t}\\ \mathbf{if}\;y \leq -4.2 \cdot 10^{+173}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+181}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (log (* z y)) t))))
   (if (<= y -4.2e+173)
     t_1
     (if (<= y 3.6e+181) (- x (/ (* (expm1 z) y) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (log((z * y)) / t);
	double tmp;
	if (y <= -4.2e+173) {
		tmp = t_1;
	} else if (y <= 3.6e+181) {
		tmp = x - ((expm1(z) * y) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x - (Math.log((z * y)) / t);
	double tmp;
	if (y <= -4.2e+173) {
		tmp = t_1;
	} else if (y <= 3.6e+181) {
		tmp = x - ((Math.expm1(z) * y) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (math.log((z * y)) / t)
	tmp = 0
	if y <= -4.2e+173:
		tmp = t_1
	elif y <= 3.6e+181:
		tmp = x - ((math.expm1(z) * y) / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(log(Float64(z * y)) / t))
	tmp = 0.0
	if (y <= -4.2e+173)
		tmp = t_1;
	elseif (y <= 3.6e+181)
		tmp = Float64(x - Float64(Float64(expm1(z) * y) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.2e+173], t$95$1, If[LessEqual[y, 3.6e+181], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.2e173 or 3.59999999999999985e181 < y

   1. Initial program 35.4%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto x - \frac{\log \left(z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right) + \color{blue}{1}\right)}{t} \]

      2. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right) \cdot z + 1\right)}{t} \]

      3. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right), \color{blue}{z}, 1\right)\right)}{t} \]

      4. +-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right) + y, z, 1\right)\right)}{t} \]

      5. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right) \cdot z + y, z, 1\right)\right)}{t} \]

      6. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y, z, y\right), z, 1\right)\right)}{t} \]

      7. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, y \cdot z, \frac{1}{2} \cdot y\right), z, y\right), z, 1\right)\right)}{t} \]

      8. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, z \cdot y, \frac{1}{2} \cdot y\right), z, y\right), z, 1\right)\right)}{t} \]

      9. lower-*.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, z \cdot y, \frac{1}{2} \cdot y\right), z, y\right), z, 1\right)\right)}{t} \]

      10. lower-*.f64 63.1

          \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, 0.5 \cdot y\right), z, y\right), z, 1\right)\right)}{t} \]

   4. Applied rewrites 63.1%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, 0.5 \cdot y\right), z, y\right), z, 1\right)\right)}}{t} \]

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\left(z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot z\right)\right)\right) \cdot y\right)}{t} \]

      2. lower-*.f64 N/A

         \[\leadsto x - \frac{\log \left(\left(z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot z\right)\right)\right) \cdot y\right)}{t} \]

      3. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\left(\left(1 + z \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot z\right)\right) \cdot z\right) \cdot y\right)}{t} \]

      4. lower-*.f64 N/A

         \[\leadsto x - \frac{\log \left(\left(\left(1 + z \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot z\right)\right) \cdot z\right) \cdot y\right)}{t} \]

      5. +-commutative N/A

         \[\leadsto x - \frac{\log \left(\left(\left(z \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot z\right) + 1\right) \cdot z\right) \cdot y\right)}{t} \]

      6. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\left(\left(\left(\frac{1}{2} + \frac{1}{6} \cdot z\right) \cdot z + 1\right) \cdot z\right) \cdot y\right)}{t} \]

      7. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \left(\left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot z, z, 1\right) \cdot z\right) \cdot y\right)}{t} \]

      8. +-commutative N/A

         \[\leadsto x - \frac{\log \left(\left(\mathsf{fma}\left(\frac{1}{6} \cdot z + \frac{1}{2}, z, 1\right) \cdot z\right) \cdot y\right)}{t} \]

      9. lower-fma.f64 55.1

         \[\leadsto x - \frac{\log \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right), z, 1\right) \cdot z\right) \cdot y\right)}{t} \]

   7. Applied rewrites 55.1%

      \[\leadsto x - \frac{\log \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right), z, 1\right) \cdot z\right) \cdot \color{blue}{y}\right)}{t} \]

   8. Taylor expanded in z around 0

      \[\leadsto x - \frac{\log \left(z \cdot y\right)}{t} \]
   9. Step-by-step derivation

   10. Applied rewrites 57.5%

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

   if -4.2e173 < y < 3.59999999999999985e181

   1. Initial program 66.4%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 91.7

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

   4. Applied rewrites 91.7%

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

Alternative 9: 85.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+181}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right) \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.15e+181) x (- x (/ (* (expm1 z) y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.15e+181) {
		tmp = x;
	} else {
		tmp = x - ((expm1(z) * y) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.15e+181) {
		tmp = x;
	} else {
		tmp = x - ((Math.expm1(z) * y) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1.15e+181:
		tmp = x
	else:
		tmp = x - ((math.expm1(z) * y) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.15e+181)
		tmp = x;
	else
		tmp = Float64(x - Float64(Float64(expm1(z) * y) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.15e+181], x, N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1499999999999999e181

   1. Initial program 49.7%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.1%

      \[\leadsto \color{blue}{x} \]

   if -1.1499999999999999e181 < y

   1. Initial program 63.1%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 89.4

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

   4. Applied rewrites 89.4%

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

Alternative 10: 81.2% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-69}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, \frac{z}{t}, -x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.2e-69) x (- (fma y (/ z t) (- x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.2e-69) {
		tmp = x;
	} else {
		tmp = -fma(y, (z / t), -x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.2e-69)
		tmp = x;
	else
		tmp = Float64(-fma(y, Float64(z / t), Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.2e-69], x, (-N[(y * N[(z / t), $MachinePrecision] + (-x)), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if z < -1.2000000000000001e-69

   1. Initial program 77.6%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 64.3%

      \[\leadsto \color{blue}{x} \]

   if -1.2000000000000001e-69 < z

   1. Initial program 52.8%

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

   2. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 73.7%

      \[\leadsto \color{blue}{-\left(\frac{\mathsf{log1p}\left(e^{z} \cdot y - y\right)}{t \cdot x} - 1\right) \cdot x} \]

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 90.9

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

   7. Applied rewrites 90.9%

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

Alternative 11: 80.1% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-69}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.2e-69) x (- x (/ (* z y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.2e-69) {
		tmp = x;
	} else {
		tmp = x - ((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) :: tmp
    if (z <= (-1.2d-69)) then
        tmp = x
    else
        tmp = x - ((z * y) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.2e-69) {
		tmp = x;
	} else {
		tmp = x - ((z * y) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.2e-69:
		tmp = x
	else:
		tmp = x - ((z * y) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.2e-69)
		tmp = x;
	else
		tmp = Float64(x - Float64(Float64(z * y) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.2e-69)
		tmp = x;
	else
		tmp = x - ((z * y) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.2e-69], x, N[(x - N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2000000000000001e-69

   1. Initial program 77.6%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 64.3%

      \[\leadsto \color{blue}{x} \]

   if -1.2000000000000001e-69 < z

   1. Initial program 52.8%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 89.2

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

   4. Applied rewrites 89.2%

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

Alternative 12: 78.1% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-69}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-y}{t}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.2e-69) x (fma (/ (- y) t) z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.2e-69) {
		tmp = x;
	} else {
		tmp = fma((-y / t), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.2e-69)
		tmp = x;
	else
		tmp = fma(Float64(Float64(-y) / t), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.2e-69], x, N[(N[((-y) / t), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2000000000000001e-69

   1. Initial program 77.6%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 64.3%

      \[\leadsto \color{blue}{x} \]

   if -1.2000000000000001e-69 < z

   1. Initial program 52.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 78.3%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 86.1

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

   7. Applied rewrites 86.1%

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

Alternative 13: 71.2% accurate, 226.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

function code(x, y, z, t)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := x

Derivation

1. Initial program 61.8%

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

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 71.2%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Developer Target 1: 74.5% 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 2025095 
(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)))