Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2

Percentage Accurate: 77.9% → 99.5%

Time: 12.9s

Alternatives: 9

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-5d-310)) then
        tmp = (x * (log(-x) - log(-y))) - z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5e-310:
		tmp = (x * (math.log(-x) - math.log(-y))) - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-310)
		tmp = (x * (log(-x) - log(-y))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e-310], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 83.2%

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

   3. Taylor expanded in y around -inf 99.3%

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

      1. metadata-eval 99.3%

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

      2. distribute-neg-frac 99.3%

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

      3. distribute-frac-neg2 99.3%

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

      4. neg-mul-1 99.3%

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

      5. log-rec 99.3%

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

      6. sub-neg 99.3%

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

   5. Simplified 99.3%

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

   if -4.999999999999985e-310 < y

   1. Initial program 77.0%

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

   3. Taylor expanded in x around 0 99.5%

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

      1. log-rec 99.5%

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

      2. sub-neg 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 5 \cdot 10^{+293}\right):\\ \;\;\;\;z \cdot \left(\frac{x}{z} \cdot \log \left(y \cdot x\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, t\_0, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (* x t_0)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e+293)))
     (* z (+ (* (/ x z) (log (* y x))) -1.0))
     (fma x t_0 (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y));
	double t_1 = x * t_0;
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e+293)) {
		tmp = z * (((x / z) * log((y * x))) + -1.0);
	} else {
		tmp = fma(x, t_0, -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(x / y))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e+293))
		tmp = Float64(z * Float64(Float64(Float64(x / z) * log(Float64(y * x))) + -1.0));
	else
		tmp = fma(x, t_0, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e+293]], $MachinePrecision]], N[(z * N[(N[(N[(x / z), $MachinePrecision] * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(x * t$95$0 + (-z)), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 5.00000000000000033e293 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 5.9%

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

   3. Taylor expanded in z around inf 5.9%

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

      1. sub-neg 5.9%

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

      2. associate-/l* 5.9%

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

      3. metadata-eval 5.9%

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

   5. Simplified 5.9%

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

      1. clear-num 5.9%

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

      2. un-div-inv 5.9%

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

      3. log-div 48.7%

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

      4. sub-neg 48.7%

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

      5. add-log-exp 48.7%

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

      6. sum-log 4.2%

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

      7. add-sqr-sqrt 2.0%

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

      8. sqrt-unprod 23.4%

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

      9. sqr-neg 23.4%

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

      10. sqrt-unprod 21.5%

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

      11. add-sqr-sqrt 32.2%

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

      12. add-exp-log 53.7%

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

   7. Applied egg-rr 53.7%

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

      1. associate-/r/ 53.7%

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

   9. Simplified 53.7%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.00000000000000033e293

   1. Initial program 99.8%

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

      1. fma-neg 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+293}\right):\\ \;\;\;\;z \cdot \left(\frac{x}{z} \cdot \log \left(y \cdot x\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 5 \cdot 10^{+293}\right):\\ \;\;\;\;z \cdot \left(\frac{x}{z} \cdot \log \left(y \cdot x\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e+293)))
     (* z (+ (* (/ x z) (log (* y x))) -1.0))
     (- t_0 z))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 5e+293)) {
		tmp = z * (((x / z) * log((y * x))) + -1.0);
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 5e+293)) {
		tmp = z * (((x / z) * Math.log((y * x))) + -1.0);
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 5e+293):
		tmp = z * (((x / z) * math.log((y * x))) + -1.0)
	else:
		tmp = t_0 - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 5e+293))
		tmp = Float64(z * Float64(Float64(Float64(x / z) * log(Float64(y * x))) + -1.0));
	else
		tmp = Float64(t_0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 5e+293)))
		tmp = z * (((x / z) * log((y * x))) + -1.0);
	else
		tmp = t_0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 5e+293]], $MachinePrecision]], N[(z * N[(N[(N[(x / z), $MachinePrecision] * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 5.00000000000000033e293 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 5.9%

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

   3. Taylor expanded in z around inf 5.9%

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

      1. sub-neg 5.9%

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

      2. associate-/l* 5.9%

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

      3. metadata-eval 5.9%

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

   5. Simplified 5.9%

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

      1. clear-num 5.9%

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

      2. un-div-inv 5.9%

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

      3. log-div 48.7%

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

      4. sub-neg 48.7%

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

      5. add-log-exp 48.7%

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

      6. sum-log 4.2%

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

      7. add-sqr-sqrt 2.0%

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

      8. sqrt-unprod 23.4%

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

      9. sqr-neg 23.4%

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

      10. sqrt-unprod 21.5%

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

      11. add-sqr-sqrt 32.2%

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

      12. add-exp-log 53.7%

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

   7. Applied egg-rr 53.7%

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

      1. associate-/r/ 53.7%

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

   9. Simplified 53.7%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.00000000000000033e293

   1. Initial program 99.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+293}\right):\\ \;\;\;\;z \cdot \left(\frac{x}{z} \cdot \log \left(y \cdot x\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 5 \cdot 10^{+293}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e+293))) (- z) (- t_0 z))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 5e+293)) {
		tmp = -z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 5e+293)) {
		tmp = -z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 5e+293):
		tmp = -z
	else:
		tmp = t_0 - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 5e+293))
		tmp = Float64(-z);
	else
		tmp = Float64(t_0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 5e+293)))
		tmp = -z;
	else
		tmp = t_0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 5e+293]], $MachinePrecision]], (-z), N[(t$95$0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 5.00000000000000033e293 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 5.9%

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

   3. Taylor expanded in x around 0 50.9%

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

      1. mul-1-neg 50.9%

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

   5. Simplified 50.9%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.00000000000000033e293

   1. Initial program 99.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+293}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-178}:\\ \;\;\;\;\left(-z\right) - x \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6.5e-178)
   (- (- z) (* x (log (/ y x))))
   (if (<= x -1e-308) (- z) (- (* x (- (log x) (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.5e-178) {
		tmp = -z - (x * log((y / x)));
	} else if (x <= -1e-308) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-6.5d-178)) then
        tmp = -z - (x * log((y / x)))
    else if (x <= (-1d-308)) then
        tmp = -z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.5e-178) {
		tmp = -z - (x * Math.log((y / x)));
	} else if (x <= -1e-308) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -6.5e-178:
		tmp = -z - (x * math.log((y / x)))
	elif x <= -1e-308:
		tmp = -z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6.5e-178)
		tmp = Float64(Float64(-z) - Float64(x * log(Float64(y / x))));
	elseif (x <= -1e-308)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -6.5e-178)
		tmp = -z - (x * log((y / x)));
	elseif (x <= -1e-308)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6.5e-178], N[((-z) - N[(x * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-308], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5000000000000002e-178

   1. Initial program 87.7%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 87.7%

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

      2. log-div 88.8%

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

      3. metadata-eval 88.8%

         \[\leadsto x \cdot \left(\color{blue}{0} - \log \left(\frac{y}{x}\right)\right) - z \]

   4. Applied egg-rr 88.8%

      \[\leadsto x \cdot \color{blue}{\left(0 - \log \left(\frac{y}{x}\right)\right)} - z \]
   5. Step-by-step derivation

      1. neg-sub0 88.8%

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

   6. Simplified 88.8%

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

   if -6.5000000000000002e-178 < x < -9.9999999999999991e-309

   1. Initial program 69.8%

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

   3. Taylor expanded in x around 0 82.5%

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

      1. mul-1-neg 82.5%

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

   5. Simplified 82.5%

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

   if -9.9999999999999991e-309 < x

   1. Initial program 77.0%

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

   3. Taylor expanded in x around 0 99.5%

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

      1. log-rec 99.5%

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

      2. sub-neg 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-178}:\\ \;\;\;\;\left(-z\right) - x \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.015:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{-139}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-188} \lor \neg \left(z \leq 8.5 \cdot 10^{-59}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\log \left(\frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.015)
   (- z)
   (if (<= z -1.05e-139)
     (* x (log (/ x y)))
     (if (or (<= z -1.3e-188) (not (<= z 8.5e-59)))
       (- z)
       (* x (- (log (/ y x))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.015) {
		tmp = -z;
	} else if (z <= -1.05e-139) {
		tmp = x * log((x / y));
	} else if ((z <= -1.3e-188) || !(z <= 8.5e-59)) {
		tmp = -z;
	} else {
		tmp = x * -log((y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-0.015d0)) then
        tmp = -z
    else if (z <= (-1.05d-139)) then
        tmp = x * log((x / y))
    else if ((z <= (-1.3d-188)) .or. (.not. (z <= 8.5d-59))) then
        tmp = -z
    else
        tmp = x * -log((y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.015) {
		tmp = -z;
	} else if (z <= -1.05e-139) {
		tmp = x * Math.log((x / y));
	} else if ((z <= -1.3e-188) || !(z <= 8.5e-59)) {
		tmp = -z;
	} else {
		tmp = x * -Math.log((y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -0.015:
		tmp = -z
	elif z <= -1.05e-139:
		tmp = x * math.log((x / y))
	elif (z <= -1.3e-188) or not (z <= 8.5e-59):
		tmp = -z
	else:
		tmp = x * -math.log((y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.015)
		tmp = Float64(-z);
	elseif (z <= -1.05e-139)
		tmp = Float64(x * log(Float64(x / y)));
	elseif ((z <= -1.3e-188) || !(z <= 8.5e-59))
		tmp = Float64(-z);
	else
		tmp = Float64(x * Float64(-log(Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -0.015)
		tmp = -z;
	elseif (z <= -1.05e-139)
		tmp = x * log((x / y));
	elseif ((z <= -1.3e-188) || ~((z <= 8.5e-59)))
		tmp = -z;
	else
		tmp = x * -log((y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.015], (-z), If[LessEqual[z, -1.05e-139], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, -1.3e-188], N[Not[LessEqual[z, 8.5e-59]], $MachinePrecision]], (-z), N[(x * (-N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.014999999999999999 or -1.05000000000000004e-139 < z < -1.3e-188 or 8.49999999999999933e-59 < z

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0 77.2%

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

      1. mul-1-neg 77.2%

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

   5. Simplified 77.2%

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

   if -0.014999999999999999 < z < -1.05000000000000004e-139

   1. Initial program 77.3%

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

   3. Taylor expanded in z around 0 62.8%

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

   if -1.3e-188 < z < 8.49999999999999933e-59

   1. Initial program 81.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 53.5%

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

      2. clear-num 53.4%

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

      3. fma-define 53.4%

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

      4. difference-of-squares 53.4%

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

      5. add-sqr-sqrt 37.9%

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

      6. sqrt-unprod 53.4%

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

      7. sqr-neg 53.4%

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

      8. sqrt-unprod 15.5%

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

      9. add-sqr-sqrt 50.9%

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

      10. sub-neg 50.9%

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

      11. pow2 50.9%

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

      12. fma-neg 50.9%

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

      13. add-sqr-sqrt 15.5%

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

      14. sqrt-unprod 50.9%

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

      15. sqr-neg 50.9%

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

      16. sqrt-unprod 35.5%

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

      17. add-sqr-sqrt 50.9%

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

   4. Applied egg-rr 50.9%

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

   5. Taylor expanded in z around 0 72.8%

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

      1. associate-/r* 72.6%

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

   7. Simplified 72.6%

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

      1. frac-2neg 72.6%

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

      2. metadata-eval 72.6%

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

      3. div-inv 72.6%

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

      4. distribute-neg-frac2 72.6%

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

      5. neg-log 72.6%

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

      6. clear-num 73.6%

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

   9. Applied egg-rr 73.6%

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

       1. mul-1-neg 73.6%

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

       2. associate-/l/ 73.8%

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

       3. remove-double-div 73.9%

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

       4. distribute-rgt-neg-out 73.9%

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

   11. Simplified 73.9%

       \[\leadsto \color{blue}{\log \left(\frac{y}{x}\right) \cdot \left(-x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.015:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{-139}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-188} \lor \neg \left(z \leq 8.5 \cdot 10^{-59}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\log \left(\frac{y}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.045 \lor \neg \left(z \leq -8.5 \cdot 10^{-140}\right) \land \left(z \leq -1.3 \cdot 10^{-188} \lor \neg \left(z \leq 3.9 \cdot 10^{-59}\right)\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -0.045)
         (and (not (<= z -8.5e-140))
              (or (<= z -1.3e-188) (not (<= z 3.9e-59)))))
   (- z)
   (* x (log (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.045) || (!(z <= -8.5e-140) && ((z <= -1.3e-188) || !(z <= 3.9e-59)))) {
		tmp = -z;
	} else {
		tmp = x * log((x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-0.045d0)) .or. (.not. (z <= (-8.5d-140))) .and. (z <= (-1.3d-188)) .or. (.not. (z <= 3.9d-59))) then
        tmp = -z
    else
        tmp = x * log((x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.045) || (!(z <= -8.5e-140) && ((z <= -1.3e-188) || !(z <= 3.9e-59)))) {
		tmp = -z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -0.045) or (not (z <= -8.5e-140) and ((z <= -1.3e-188) or not (z <= 3.9e-59))):
		tmp = -z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -0.045) || (!(z <= -8.5e-140) && ((z <= -1.3e-188) || !(z <= 3.9e-59))))
		tmp = Float64(-z);
	else
		tmp = Float64(x * log(Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -0.045) || (~((z <= -8.5e-140)) && ((z <= -1.3e-188) || ~((z <= 3.9e-59)))))
		tmp = -z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -0.045], And[N[Not[LessEqual[z, -8.5e-140]], $MachinePrecision], Or[LessEqual[z, -1.3e-188], N[Not[LessEqual[z, 3.9e-59]], $MachinePrecision]]]], (-z), N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.044999999999999998 or -8.49999999999999997e-140 < z < -1.3e-188 or 3.90000000000000019e-59 < z

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0 77.2%

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

      1. mul-1-neg 77.2%

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

   5. Simplified 77.2%

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

   if -0.044999999999999998 < z < -8.49999999999999997e-140 or -1.3e-188 < z < 3.90000000000000019e-59

   1. Initial program 80.3%

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

   3. Taylor expanded in z around 0 70.3%

      \[\leadsto \color{blue}{x \cdot \log \left(\frac{x}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.045 \lor \neg \left(z \leq -8.5 \cdot 10^{-140}\right) \land \left(z \leq -1.3 \cdot 10^{-188} \lor \neg \left(z \leq 3.9 \cdot 10^{-59}\right)\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.8% accurate, 53.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

3. Taylor expanded in x around 0 54.1%

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

   1. mul-1-neg 54.1%

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

5. Simplified 54.1%

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

6. Final simplification 54.1%

   \[\leadsto -z \]
7. Add Preprocessing

Alternative 9: 2.3% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 80.0%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-- 41.7%

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

   2. clear-num 41.6%

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

   3. fma-define 41.6%

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

   4. difference-of-squares 42.2%

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

   5. add-sqr-sqrt 22.6%

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

   6. sqrt-unprod 31.5%

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

   7. sqr-neg 31.5%

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

   8. sqrt-unprod 8.8%

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

   9. add-sqr-sqrt 21.2%

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

   10. sub-neg 21.2%

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

   11. pow2 21.2%

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

   12. fma-neg 21.2%

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

   13. add-sqr-sqrt 8.8%

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

   14. sqrt-unprod 21.2%

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

   15. sqr-neg 21.2%

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

   16. sqrt-unprod 12.3%

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

   17. add-sqr-sqrt 21.2%

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

4. Applied egg-rr 21.2%

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

5. Taylor expanded in x around 0 2.2%

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

6. Final simplification 2.2%

   \[\leadsto z \]
7. Add Preprocessing

Developer target: 88.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y < 7.595077799083773 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< y 7.595077799083773e-308)
   (- (* x (log (/ x y))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y < 7.595077799083773e-308) {
		tmp = (x * log((x / y))) - z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y < 7.595077799083773d-308) then
        tmp = (x * log((x / y))) - z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y < 7.595077799083773e-308) {
		tmp = (x * Math.log((x / y))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y < 7.595077799083773e-308:
		tmp = (x * math.log((x / y))) - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y < 7.595077799083773e-308)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y < 7.595077799083773e-308)
		tmp = (x * log((x / y))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[y, 7.595077799083773e-308], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Reproduce

herbie shell --seed 2024067 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (if (< y 7.595077799083773e-308) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z))

  (- (* x (log (/ x y))) z))