Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A

Percentage Accurate: 99.9% → 99.9%

Time: 13.6s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]

2. Final simplification 99.9%

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

Alternative 2: 89.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t_1 - y\\ \mathbf{if}\;t_2 \leq -6 \cdot 10^{+155}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)))
   (if (<= t_2 -6e+155)
     t_2
     (if (<= t_2 4e-12) (- (log t) (+ y z)) (- t_1 z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -6e+155) {
		tmp = t_2;
	} else if (t_2 <= 4e-12) {
		tmp = log(t) - (y + z);
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = t_1 - y
    if (t_2 <= (-6d+155)) then
        tmp = t_2
    else if (t_2 <= 4d-12) then
        tmp = log(t) - (y + z)
    else
        tmp = t_1 - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -6e+155) {
		tmp = t_2;
	} else if (t_2 <= 4e-12) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = t_1 - y
	tmp = 0
	if t_2 <= -6e+155:
		tmp = t_2
	elif t_2 <= 4e-12:
		tmp = math.log(t) - (y + z)
	else:
		tmp = t_1 - z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 - y)
	tmp = 0.0
	if (t_2 <= -6e+155)
		tmp = t_2;
	elseif (t_2 <= 4e-12)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = Float64(t_1 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = t_1 - y;
	tmp = 0.0;
	if (t_2 <= -6e+155)
		tmp = t_2;
	elseif (t_2 <= 4e-12)
		tmp = log(t) - (y + z);
	else
		tmp = t_1 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - y), $MachinePrecision]}, If[LessEqual[t$95$2, -6e+155], t$95$2, If[LessEqual[t$95$2, 4e-12], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -6.0000000000000003e155

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.9%

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

      2. fma-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 99.9%

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

   5. Taylor expanded in z around 0 94.0%

      \[\leadsto \color{blue}{x \cdot \log y - y} \]

   if -6.0000000000000003e155 < (-.f64 (*.f64 x (log.f64 y)) y) < 3.99999999999999992e-12

   1. Initial program 100.0%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]

   2. Taylor expanded in x around 0 92.3%

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

   if 3.99999999999999992e-12 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.6%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.6%

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

      2. fma-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in z around inf 98.4%

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

   5. Taylor expanded in y around 0 95.5%

      \[\leadsto \color{blue}{x \cdot \log y - z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -6 \cdot 10^{+155}:\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{elif}\;x \cdot \log y - y \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - z\\ \end{array} \]

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -30500000 \lor \neg \left(x \leq 4.1 \cdot 10^{-10}\right):\\ \;\;\;\;\mathsf{fma}\left(x, \log y, -y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -30500000.0) (not (<= x 4.1e-10)))
   (- (fma x (log y) (- y)) z)
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -30500000.0) || !(x <= 4.1e-10)) {
		tmp = fma(x, log(y), -y) - z;
	} else {
		tmp = log(t) - (y + z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -30500000.0) || !(x <= 4.1e-10))
		tmp = Float64(fma(x, log(y), Float64(-y)) - z);
	else
		tmp = Float64(log(t) - Float64(y + z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -30500000.0], N[Not[LessEqual[x, 4.1e-10]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision] + (-y)), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.05e7 or 4.0999999999999998e-10 < x

   1. Initial program 99.7%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.7%

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

      2. fma-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around inf 98.4%

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

   if -3.05e7 < x < 4.0999999999999998e-10

   1. Initial program 100.0%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]

   2. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -30500000 \lor \neg \left(x \leq 4.1 \cdot 10^{-10}\right):\\ \;\;\;\;\mathsf{fma}\left(x, \log y, -y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 4: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-18}:\\ \;\;\;\;\log t + \left(x \cdot \log y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, -y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.85e-18)
   (+ (log t) (- (* x (log y)) z))
   (- (fma x (log y) (- y)) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.85e-18) {
		tmp = log(t) + ((x * log(y)) - z);
	} else {
		tmp = fma(x, log(y), -y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.85e-18)
		tmp = Float64(log(t) + Float64(Float64(x * log(y)) - z));
	else
		tmp = Float64(fma(x, log(y), Float64(-y)) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.85e-18], N[(N[Log[t], $MachinePrecision] + N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Log[y], $MachinePrecision] + (-y)), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8500000000000002e-18

   1. Initial program 99.8%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]

   2. Taylor expanded in y around 0 99.8%

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

   if 1.8500000000000002e-18 < y

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.9%

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

      2. fma-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 97.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-18}:\\ \;\;\;\;\log t + \left(x \cdot \log y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, -y\right) - z\\ \end{array} \]

Alternative 5: 60.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \log t - z\\ \mathbf{if}\;x \leq -5.4 \cdot 10^{+106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-275}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+47}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- (log t) z)))
   (if (<= x -5.4e+106)
     t_1
     (if (<= x -2e+60)
       t_2
       (if (<= x -8.6e+25)
         t_1
         (if (<= x -1.7e-275) (- (log t) y) (if (<= x 8e+47) t_2 t_1)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = log(t) - z;
	double tmp;
	if (x <= -5.4e+106) {
		tmp = t_1;
	} else if (x <= -2e+60) {
		tmp = t_2;
	} else if (x <= -8.6e+25) {
		tmp = t_1;
	} else if (x <= -1.7e-275) {
		tmp = log(t) - y;
	} else if (x <= 8e+47) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = log(t) - z
    if (x <= (-5.4d+106)) then
        tmp = t_1
    else if (x <= (-2d+60)) then
        tmp = t_2
    else if (x <= (-8.6d+25)) then
        tmp = t_1
    else if (x <= (-1.7d-275)) then
        tmp = log(t) - y
    else if (x <= 8d+47) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double t_2 = Math.log(t) - z;
	double tmp;
	if (x <= -5.4e+106) {
		tmp = t_1;
	} else if (x <= -2e+60) {
		tmp = t_2;
	} else if (x <= -8.6e+25) {
		tmp = t_1;
	} else if (x <= -1.7e-275) {
		tmp = Math.log(t) - y;
	} else if (x <= 8e+47) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = math.log(t) - z
	tmp = 0
	if x <= -5.4e+106:
		tmp = t_1
	elif x <= -2e+60:
		tmp = t_2
	elif x <= -8.6e+25:
		tmp = t_1
	elif x <= -1.7e-275:
		tmp = math.log(t) - y
	elif x <= 8e+47:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(log(t) - z)
	tmp = 0.0
	if (x <= -5.4e+106)
		tmp = t_1;
	elseif (x <= -2e+60)
		tmp = t_2;
	elseif (x <= -8.6e+25)
		tmp = t_1;
	elseif (x <= -1.7e-275)
		tmp = Float64(log(t) - y);
	elseif (x <= 8e+47)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = log(t) - z;
	tmp = 0.0;
	if (x <= -5.4e+106)
		tmp = t_1;
	elseif (x <= -2e+60)
		tmp = t_2;
	elseif (x <= -8.6e+25)
		tmp = t_1;
	elseif (x <= -1.7e-275)
		tmp = log(t) - y;
	elseif (x <= 8e+47)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[x, -5.4e+106], t$95$1, If[LessEqual[x, -2e+60], t$95$2, If[LessEqual[x, -8.6e+25], t$95$1, If[LessEqual[x, -1.7e-275], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision], If[LessEqual[x, 8e+47], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.40000000000000012e106 or -1.9999999999999999e60 < x < -8.59999999999999996e25 or 8.0000000000000004e47 < x

   1. Initial program 99.7%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.7%

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

      2. fma-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around inf 99.7%

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

   5. Taylor expanded in x around inf 65.9%

      \[\leadsto \color{blue}{x \cdot \log y} \]

   if -5.40000000000000012e106 < x < -1.9999999999999999e60 or -1.69999999999999984e-275 < x < 8.0000000000000004e47

   1. Initial program 100.0%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]

   2. Taylor expanded in z around inf 63.3%

      \[\leadsto \color{blue}{-1 \cdot z} + \log t \]
   3. Step-by-step derivation

      1. mul-1-neg 63.3%

         \[\leadsto \color{blue}{\left(-z\right)} + \log t \]

   4. Simplified 63.3%

      \[\leadsto \color{blue}{\left(-z\right)} + \log t \]

   if -8.59999999999999996e25 < x < -1.69999999999999984e-275

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]

   2. Taylor expanded in y around inf 72.4%

      \[\leadsto \color{blue}{-1 \cdot y} + \log t \]
   3. Step-by-step derivation

      1. mul-1-neg 72.4%

         \[\leadsto \color{blue}{\left(-y\right)} + \log t \]

   4. Simplified 72.4%

      \[\leadsto \color{blue}{\left(-y\right)} + \log t \]

   5. Taylor expanded in y around 0 72.4%

      \[\leadsto \color{blue}{\log t + -1 \cdot y} \]

   7. Simplified 72.4%

      \[\leadsto \color{blue}{\log t - y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+106}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -2 \cdot 10^{+60}:\\ \;\;\;\;\log t - z\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-275}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+47}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]

Alternative 6: 89.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{+114} \lor \neg \left(x \leq -1.3 \cdot 10^{+64} \lor \neg \left(x \leq -1000000000\right) \land x \leq 6.2 \cdot 10^{+47}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.3e+114)
         (not
          (or (<= x -1.3e+64)
              (and (not (<= x -1000000000.0)) (<= x 6.2e+47)))))
   (- (* x (log y)) y)
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.3e+114) || !((x <= -1.3e+64) || (!(x <= -1000000000.0) && (x <= 6.2e+47)))) {
		tmp = (x * log(y)) - y;
	} else {
		tmp = log(t) - (y + z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-4.3d+114)) .or. (.not. (x <= (-1.3d+64)) .or. (.not. (x <= (-1000000000.0d0))) .and. (x <= 6.2d+47))) then
        tmp = (x * log(y)) - y
    else
        tmp = log(t) - (y + z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.3e+114) || !((x <= -1.3e+64) || (!(x <= -1000000000.0) && (x <= 6.2e+47)))) {
		tmp = (x * Math.log(y)) - y;
	} else {
		tmp = Math.log(t) - (y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -4.3e+114) or not ((x <= -1.3e+64) or (not (x <= -1000000000.0) and (x <= 6.2e+47))):
		tmp = (x * math.log(y)) - y
	else:
		tmp = math.log(t) - (y + z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.3e+114) || !((x <= -1.3e+64) || (!(x <= -1000000000.0) && (x <= 6.2e+47))))
		tmp = Float64(Float64(x * log(y)) - y);
	else
		tmp = Float64(log(t) - Float64(y + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -4.3e+114) || ~(((x <= -1.3e+64) || (~((x <= -1000000000.0)) && (x <= 6.2e+47)))))
		tmp = (x * log(y)) - y;
	else
		tmp = log(t) - (y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.3e+114], N[Not[Or[LessEqual[x, -1.3e+64], And[N[Not[LessEqual[x, -1000000000.0]], $MachinePrecision], LessEqual[x, 6.2e+47]]]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.3000000000000001e114 or -1.29999999999999998e64 < x < -1e9 or 6.2000000000000001e47 < x

   1. Initial program 99.7%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.7%

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

      2. fma-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around inf 99.0%

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

   5. Taylor expanded in z around 0 86.4%

      \[\leadsto \color{blue}{x \cdot \log y - y} \]

   if -4.3000000000000001e114 < x < -1.29999999999999998e64 or -1e9 < x < 6.2000000000000001e47

   1. Initial program 100.0%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]

   2. Taylor expanded in x around 0 96.6%

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{+114} \lor \neg \left(x \leq -1.3 \cdot 10^{+64} \lor \neg \left(x \leq -1000000000\right) \land x \leq 6.2 \cdot 10^{+47}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 7: 60.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t - y\\ \mathbf{if}\;z \leq -2.15 \cdot 10^{+114}:\\ \;\;\;\;y - z\\ \mathbf{elif}\;z \leq -8.8 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (log t) y)))
   (if (<= z -2.15e+114)
     (- y z)
     (if (<= z -8.8e+26)
       t_1
       (if (<= z -7e-16) (* x (log y)) (if (<= z 1.6e+53) t_1 (- z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(t) - y;
	double tmp;
	if (z <= -2.15e+114) {
		tmp = y - z;
	} else if (z <= -8.8e+26) {
		tmp = t_1;
	} else if (z <= -7e-16) {
		tmp = x * log(y);
	} else if (z <= 1.6e+53) {
		tmp = t_1;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    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 = log(t) - y
    if (z <= (-2.15d+114)) then
        tmp = y - z
    else if (z <= (-8.8d+26)) then
        tmp = t_1
    else if (z <= (-7d-16)) then
        tmp = x * log(y)
    else if (z <= 1.6d+53) then
        tmp = t_1
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(t) - y;
	double tmp;
	if (z <= -2.15e+114) {
		tmp = y - z;
	} else if (z <= -8.8e+26) {
		tmp = t_1;
	} else if (z <= -7e-16) {
		tmp = x * Math.log(y);
	} else if (z <= 1.6e+53) {
		tmp = t_1;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(t) - y
	tmp = 0
	if z <= -2.15e+114:
		tmp = y - z
	elif z <= -8.8e+26:
		tmp = t_1
	elif z <= -7e-16:
		tmp = x * math.log(y)
	elif z <= 1.6e+53:
		tmp = t_1
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(t) - y)
	tmp = 0.0
	if (z <= -2.15e+114)
		tmp = Float64(y - z);
	elseif (z <= -8.8e+26)
		tmp = t_1;
	elseif (z <= -7e-16)
		tmp = Float64(x * log(y));
	elseif (z <= 1.6e+53)
		tmp = t_1;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(t) - y;
	tmp = 0.0;
	if (z <= -2.15e+114)
		tmp = y - z;
	elseif (z <= -8.8e+26)
		tmp = t_1;
	elseif (z <= -7e-16)
		tmp = x * log(y);
	elseif (z <= 1.6e+53)
		tmp = t_1;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[z, -2.15e+114], N[(y - z), $MachinePrecision], If[LessEqual[z, -8.8e+26], t$95$1, If[LessEqual[z, -7e-16], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.6e+53], t$95$1, (-z)]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.15e114

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.9%

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

      2. fma-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 99.9%

      \[\leadsto \mathsf{fma}\left(x, \log y, -y\right) - \color{blue}{z} \]
   5. Step-by-step derivation

      1. fma-neg 99.9%

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

      2. add-cube-cbrt 99.3%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x \cdot \log y - y} \cdot \sqrt[3]{x \cdot \log y - y}\right) \cdot \sqrt[3]{x \cdot \log y - y}} - z \]

      3. pow3 99.3%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \log y - y}\right)}^{3}} - z \]

      4. fma-neg 99.3%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, \log y, -y\right)}}\right)}^{3} - z \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}\right)}\right)}^{3} - z \]

      6. sqrt-unprod 67.2%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right)}\right)}^{3} - z \]

      7. sqr-neg 67.2%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \sqrt{\color{blue}{y \cdot y}}\right)}\right)}^{3} - z \]

      8. sqrt-unprod 90.8%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{\sqrt{y} \cdot \sqrt{y}}\right)}\right)}^{3} - z \]

      9. add-sqr-sqrt 90.8%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{y}\right)}\right)}^{3} - z \]

   6. Applied egg-rr 90.8%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, y\right)}\right)}^{3}} - z \]

   7. Taylor expanded in x around 0 66.9%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot y} - z \]
   8. Step-by-step derivation

      1. pow-base-1 66.9%

         \[\leadsto \color{blue}{1} \cdot y - z \]

      2. *-lft-identity 66.9%

         \[\leadsto \color{blue}{y} - z \]

   9. Simplified 66.9%

      \[\leadsto \color{blue}{y} - z \]

   if -2.15e114 < z < -8.80000000000000028e26 or -7.00000000000000035e-16 < z < 1.6e53

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]

   2. Taylor expanded in y around inf 62.3%

      \[\leadsto \color{blue}{-1 \cdot y} + \log t \]
   3. Step-by-step derivation

      1. mul-1-neg 62.3%

         \[\leadsto \color{blue}{\left(-y\right)} + \log t \]

   4. Simplified 62.3%

      \[\leadsto \color{blue}{\left(-y\right)} + \log t \]

   5. Taylor expanded in y around 0 62.3%

      \[\leadsto \color{blue}{\log t + -1 \cdot y} \]

   7. Simplified 62.3%

      \[\leadsto \color{blue}{\log t - y} \]

   if -8.80000000000000028e26 < z < -7.00000000000000035e-16

   1. Initial program 99.7%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.7%

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

      2. fma-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 91.1%

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

   5. Taylor expanded in x around inf 59.1%

      \[\leadsto \color{blue}{x \cdot \log y} \]

   if 1.6e53 < z

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.9%

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

      2. fma-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 99.9%

      \[\leadsto \mathsf{fma}\left(x, \log y, -y\right) - \color{blue}{z} \]
   5. Step-by-step derivation

      1. fma-neg 99.9%

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

      2. add-cube-cbrt 99.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x \cdot \log y - y} \cdot \sqrt[3]{x \cdot \log y - y}\right) \cdot \sqrt[3]{x \cdot \log y - y}} - z \]

      3. pow3 99.5%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \log y - y}\right)}^{3}} - z \]

      4. fma-neg 99.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, \log y, -y\right)}}\right)}^{3} - z \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}\right)}\right)}^{3} - z \]

      6. sqrt-unprod 75.2%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right)}\right)}^{3} - z \]

      7. sqr-neg 75.2%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \sqrt{\color{blue}{y \cdot y}}\right)}\right)}^{3} - z \]

      8. sqrt-unprod 82.5%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{\sqrt{y} \cdot \sqrt{y}}\right)}\right)}^{3} - z \]

      9. add-sqr-sqrt 82.5%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{y}\right)}\right)}^{3} - z \]

   6. Applied egg-rr 82.5%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, y\right)}\right)}^{3}} - z \]

   7. Taylor expanded in y around 0 83.7%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(x \cdot \log y\right)} - z \]
   8. Step-by-step derivation

      1. pow-base-1 83.7%

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

      2. *-lft-identity 83.7%

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

      3. *-commutative 83.7%

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

   9. Simplified 83.7%

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

   10. Taylor expanded in x around 0 71.1%

       \[\leadsto \color{blue}{-1 \cdot z} \]
   11. Step-by-step derivation

       1. mul-1-neg 71.1%

          \[\leadsto \color{blue}{-z} \]

   12. Simplified 71.1%

       \[\leadsto \color{blue}{-z} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+114}:\\ \;\;\;\;y - z\\ \mathbf{elif}\;z \leq -8.8 \cdot 10^{+26}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+53}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8: 84.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+115} \lor \neg \left(x \leq 3.3 \cdot 10^{+154}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -7.8e+115) (not (<= x 3.3e+154)))
   (* x (log y))
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -7.8e+115) || !(x <= 3.3e+154)) {
		tmp = x * log(y);
	} else {
		tmp = log(t) - (y + z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-7.8d+115)) .or. (.not. (x <= 3.3d+154))) then
        tmp = x * log(y)
    else
        tmp = log(t) - (y + z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -7.8e+115) || !(x <= 3.3e+154)) {
		tmp = x * Math.log(y);
	} else {
		tmp = Math.log(t) - (y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -7.8e+115) or not (x <= 3.3e+154):
		tmp = x * math.log(y)
	else:
		tmp = math.log(t) - (y + z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -7.8e+115) || !(x <= 3.3e+154))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(log(t) - Float64(y + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -7.8e+115) || ~((x <= 3.3e+154)))
		tmp = x * log(y);
	else
		tmp = log(t) - (y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -7.8e+115], N[Not[LessEqual[x, 3.3e+154]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.80000000000000012e115 or 3.3e154 < x

   1. Initial program 99.7%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.7%

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

      2. fma-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in z around inf 99.6%

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

   5. Taylor expanded in x around inf 75.4%

      \[\leadsto \color{blue}{x \cdot \log y} \]

   if -7.80000000000000012e115 < x < 3.3e154

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]

   2. Taylor expanded in x around 0 88.8%

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+115} \lor \neg \left(x \leq 3.3 \cdot 10^{+154}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 9: 49.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+24}:\\ \;\;\;\;y - z\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+43}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.2e+24) (- y z) (if (<= z 1.32e+43) (* x (log y)) (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.2e+24) {
		tmp = y - z;
	} else if (z <= 1.32e+43) {
		tmp = x * log(y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    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 <= (-2.2d+24)) then
        tmp = y - z
    else if (z <= 1.32d+43) then
        tmp = x * log(y)
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.2e+24) {
		tmp = y - z;
	} else if (z <= 1.32e+43) {
		tmp = x * Math.log(y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.2e+24:
		tmp = y - z
	elif z <= 1.32e+43:
		tmp = x * math.log(y)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.2e+24)
		tmp = Float64(y - z);
	elseif (z <= 1.32e+43)
		tmp = Float64(x * log(y));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2.2e+24)
		tmp = y - z;
	elseif (z <= 1.32e+43)
		tmp = x * log(y);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.2e+24], N[(y - z), $MachinePrecision], If[LessEqual[z, 1.32e+43], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 3 regimes

2. if z < -2.20000000000000002e24

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.9%

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

      2. fma-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 99.9%

      \[\leadsto \mathsf{fma}\left(x, \log y, -y\right) - \color{blue}{z} \]
   5. Step-by-step derivation

      1. fma-neg 99.9%

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

      2. add-cube-cbrt 99.1%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x \cdot \log y - y} \cdot \sqrt[3]{x \cdot \log y - y}\right) \cdot \sqrt[3]{x \cdot \log y - y}} - z \]

      3. pow3 99.2%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \log y - y}\right)}^{3}} - z \]

      4. fma-neg 99.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, \log y, -y\right)}}\right)}^{3} - z \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}\right)}\right)}^{3} - z \]

      6. sqrt-unprod 57.1%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right)}\right)}^{3} - z \]

      7. sqr-neg 57.1%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \sqrt{\color{blue}{y \cdot y}}\right)}\right)}^{3} - z \]

      8. sqrt-unprod 73.7%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{\sqrt{y} \cdot \sqrt{y}}\right)}\right)}^{3} - z \]

      9. add-sqr-sqrt 73.7%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{y}\right)}\right)}^{3} - z \]

   6. Applied egg-rr 73.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, y\right)}\right)}^{3}} - z \]

   7. Taylor expanded in x around 0 54.9%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot y} - z \]
   8. Step-by-step derivation

      1. pow-base-1 54.9%

         \[\leadsto \color{blue}{1} \cdot y - z \]

      2. *-lft-identity 54.9%

         \[\leadsto \color{blue}{y} - z \]

   9. Simplified 54.9%

      \[\leadsto \color{blue}{y} - z \]

   if -2.20000000000000002e24 < z < 1.32e43

   1. Initial program 99.8%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.8%

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

      2. fma-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 79.0%

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

   5. Taylor expanded in x around inf 40.3%

      \[\leadsto \color{blue}{x \cdot \log y} \]

   if 1.32e43 < z

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.9%

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

      2. fma-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 99.9%

      \[\leadsto \mathsf{fma}\left(x, \log y, -y\right) - \color{blue}{z} \]
   5. Step-by-step derivation

      1. fma-neg 99.9%

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

      2. add-cube-cbrt 99.3%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x \cdot \log y - y} \cdot \sqrt[3]{x \cdot \log y - y}\right) \cdot \sqrt[3]{x \cdot \log y - y}} - z \]

      3. pow3 99.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \log y - y}\right)}^{3}} - z \]

      4. fma-neg 99.4%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, \log y, -y\right)}}\right)}^{3} - z \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}\right)}\right)}^{3} - z \]

      6. sqrt-unprod 71.6%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right)}\right)}^{3} - z \]

      7. sqr-neg 71.6%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \sqrt{\color{blue}{y \cdot y}}\right)}\right)}^{3} - z \]

      8. sqrt-unprod 78.5%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{\sqrt{y} \cdot \sqrt{y}}\right)}\right)}^{3} - z \]

      9. add-sqr-sqrt 78.5%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{y}\right)}\right)}^{3} - z \]

   6. Applied egg-rr 78.5%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, y\right)}\right)}^{3}} - z \]

   7. Taylor expanded in y around 0 79.8%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(x \cdot \log y\right)} - z \]
   8. Step-by-step derivation

      1. pow-base-1 79.8%

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

      2. *-lft-identity 79.8%

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

      3. *-commutative 79.8%

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

   9. Simplified 79.8%

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

   10. Taylor expanded in x around 0 67.9%

       \[\leadsto \color{blue}{-1 \cdot z} \]
   11. Step-by-step derivation

       1. mul-1-neg 67.9%

          \[\leadsto \color{blue}{-z} \]

   12. Simplified 67.9%

       \[\leadsto \color{blue}{-z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+24}:\\ \;\;\;\;y - z\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+43}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 10: 42.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -185:\\ \;\;\;\;y - z\\ \mathbf{elif}\;z \leq 14.5:\\ \;\;\;\;\log t\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -185.0) (- y z) (if (<= z 14.5) (log t) (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -185.0) {
		tmp = y - z;
	} else if (z <= 14.5) {
		tmp = log(t);
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    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 <= (-185.0d0)) then
        tmp = y - z
    else if (z <= 14.5d0) then
        tmp = log(t)
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -185.0) {
		tmp = y - z;
	} else if (z <= 14.5) {
		tmp = Math.log(t);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -185.0:
		tmp = y - z
	elif z <= 14.5:
		tmp = math.log(t)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -185.0)
		tmp = Float64(y - z);
	elseif (z <= 14.5)
		tmp = log(t);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -185.0)
		tmp = y - z;
	elseif (z <= 14.5)
		tmp = log(t);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -185.0], N[(y - z), $MachinePrecision], If[LessEqual[z, 14.5], N[Log[t], $MachinePrecision], (-z)]]

Derivation

1. Split input into 3 regimes

2. if z < -185

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.9%

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

      2. fma-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 98.7%

      \[\leadsto \mathsf{fma}\left(x, \log y, -y\right) - \color{blue}{z} \]
   5. Step-by-step derivation

      1. fma-neg 98.7%

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

      2. add-cube-cbrt 97.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x \cdot \log y - y} \cdot \sqrt[3]{x \cdot \log y - y}\right) \cdot \sqrt[3]{x \cdot \log y - y}} - z \]

      3. pow3 97.7%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \log y - y}\right)}^{3}} - z \]

      4. fma-neg 97.7%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, \log y, -y\right)}}\right)}^{3} - z \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}\right)}\right)}^{3} - z \]

      6. sqrt-unprod 59.0%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right)}\right)}^{3} - z \]

      7. sqr-neg 59.0%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \sqrt{\color{blue}{y \cdot y}}\right)}\right)}^{3} - z \]

      8. sqrt-unprod 72.9%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{\sqrt{y} \cdot \sqrt{y}}\right)}\right)}^{3} - z \]

      9. add-sqr-sqrt 72.9%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{y}\right)}\right)}^{3} - z \]

   6. Applied egg-rr 72.9%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, y\right)}\right)}^{3}} - z \]

   7. Taylor expanded in x around 0 46.8%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot y} - z \]
   8. Step-by-step derivation

      1. pow-base-1 46.8%

         \[\leadsto \color{blue}{1} \cdot y - z \]

      2. *-lft-identity 46.8%

         \[\leadsto \color{blue}{y} - z \]

   9. Simplified 46.8%

      \[\leadsto \color{blue}{y} - z \]

   if -185 < z < 14.5

   1. Initial program 99.8%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]

   2. Taylor expanded in y around inf 61.7%

      \[\leadsto \color{blue}{-1 \cdot y} + \log t \]
   3. Step-by-step derivation

      1. mul-1-neg 61.7%

         \[\leadsto \color{blue}{\left(-y\right)} + \log t \]

   4. Simplified 61.7%

      \[\leadsto \color{blue}{\left(-y\right)} + \log t \]

   5. Taylor expanded in y around 0 22.6%

      \[\leadsto \color{blue}{\log t} \]

   if 14.5 < z

   1. Initial program 99.8%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Step-by-step derivation

      1. associate-+l- 99.8%

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

      2. fma-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 99.3%

      \[\leadsto \mathsf{fma}\left(x, \log y, -y\right) - \color{blue}{z} \]
   5. Step-by-step derivation

      1. fma-neg 99.3%

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

      2. add-cube-cbrt 98.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x \cdot \log y - y} \cdot \sqrt[3]{x \cdot \log y - y}\right) \cdot \sqrt[3]{x \cdot \log y - y}} - z \]

      3. pow3 98.7%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \log y - y}\right)}^{3}} - z \]

      4. fma-neg 98.7%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, \log y, -y\right)}}\right)}^{3} - z \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}\right)}\right)}^{3} - z \]

      6. sqrt-unprod 68.7%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right)}\right)}^{3} - z \]

      7. sqr-neg 68.7%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \sqrt{\color{blue}{y \cdot y}}\right)}\right)}^{3} - z \]

      8. sqrt-unprod 76.1%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{\sqrt{y} \cdot \sqrt{y}}\right)}\right)}^{3} - z \]

      9. add-sqr-sqrt 76.1%

         \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{y}\right)}\right)}^{3} - z \]

   6. Applied egg-rr 76.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, y\right)}\right)}^{3}} - z \]

   7. Taylor expanded in y around 0 77.5%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(x \cdot \log y\right)} - z \]
   8. Step-by-step derivation

      1. pow-base-1 77.5%

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

      2. *-lft-identity 77.5%

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

      3. *-commutative 77.5%

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

   9. Simplified 77.5%

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

   10. Taylor expanded in x around 0 62.2%

       \[\leadsto \color{blue}{-1 \cdot z} \]
   11. Step-by-step derivation

       1. mul-1-neg 62.2%

          \[\leadsto \color{blue}{-z} \]

   12. Simplified 62.2%

       \[\leadsto \color{blue}{-z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -185:\\ \;\;\;\;y - z\\ \mathbf{elif}\;z \leq 14.5:\\ \;\;\;\;\log t\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 11: 30.5% accurate, 104.5× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := (-z)

Derivation

1. Initial program 99.9%

   \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
2. Step-by-step derivation

   1. associate-+l- 99.8%

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

   2. fma-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in z around inf 88.2%

   \[\leadsto \mathsf{fma}\left(x, \log y, -y\right) - \color{blue}{z} \]
5. Step-by-step derivation

   1. fma-neg 88.2%

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

   2. add-cube-cbrt 87.1%

      \[\leadsto \color{blue}{\left(\sqrt[3]{x \cdot \log y - y} \cdot \sqrt[3]{x \cdot \log y - y}\right) \cdot \sqrt[3]{x \cdot \log y - y}} - z \]

   3. pow3 87.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \log y - y}\right)}^{3}} - z \]

   4. fma-neg 87.2%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x, \log y, -y\right)}}\right)}^{3} - z \]

   5. add-sqr-sqrt 0.0%

      \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}\right)}\right)}^{3} - z \]

   6. sqrt-unprod 49.9%

      \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right)}\right)}^{3} - z \]

   7. sqr-neg 49.9%

      \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \sqrt{\color{blue}{y \cdot y}}\right)}\right)}^{3} - z \]

   8. sqrt-unprod 56.0%

      \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{\sqrt{y} \cdot \sqrt{y}}\right)}\right)}^{3} - z \]

   9. add-sqr-sqrt 56.0%

      \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, \color{blue}{y}\right)}\right)}^{3} - z \]

6. Applied egg-rr 56.0%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, y\right)}\right)}^{3}} - z \]

7. Taylor expanded in y around 0 57.4%

   \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(x \cdot \log y\right)} - z \]
8. Step-by-step derivation

   1. pow-base-1 57.4%

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

   2. *-lft-identity 57.4%

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

   3. *-commutative 57.4%

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

9. Simplified 57.4%

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

10. Taylor expanded in x around 0 29.6%

    \[\leadsto \color{blue}{-1 \cdot z} \]
11. Step-by-step derivation

    1. mul-1-neg 29.6%

       \[\leadsto \color{blue}{-z} \]

12. Simplified 29.6%

    \[\leadsto \color{blue}{-z} \]

13. Final simplification 29.6%

    \[\leadsto -z \]

Reproduce

herbie shell --seed 2023301 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (- (* x (log y)) y) z) (log t)))