Harley's example

Percentage Accurate: 90.6% → 98.5%

Time: 2.3min

Alternatives: 4

Speedup: 835.0×

• could not determine a ground truth

  1. c_p = 3.904323854234024e+230
  2. c_n = 5.630615422536412e+56
  3. t = -136.690147027115
  4. s = 2.9908757622925687e+50

Specification

\[0 < c\_p \land 0 < c\_n\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{1 + e^{-t}}\\ t_2 := \frac{1}{1 + e^{-s}}\\ \frac{{t\_2}^{c\_p} \cdot {\left(1 - t\_2\right)}^{c\_n}}{{t\_1}^{c\_p} \cdot {\left(1 - t\_1\right)}^{c\_n}} \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ 1.0 (exp (- t))))) (t_2 (/ 1.0 (+ 1.0 (exp (- s))))))
   (/
    (* (pow t_2 c_p) (pow (- 1.0 t_2) c_n))
    (* (pow t_1 c_p) (pow (- 1.0 t_1) c_n)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 / (1.0 + exp(-t));
	double t_2 = 1.0 / (1.0 + exp(-s));
	return (pow(t_2, c_p) * pow((1.0 - t_2), c_n)) / (pow(t_1, c_p) * pow((1.0 - t_1), c_n));
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 1.0d0 / (1.0d0 + exp(-t))
    t_2 = 1.0d0 / (1.0d0 + exp(-s))
    code = ((t_2 ** c_p) * ((1.0d0 - t_2) ** c_n)) / ((t_1 ** c_p) * ((1.0d0 - t_1) ** c_n))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 / (1.0 + Math.exp(-t));
	double t_2 = 1.0 / (1.0 + Math.exp(-s));
	return (Math.pow(t_2, c_p) * Math.pow((1.0 - t_2), c_n)) / (Math.pow(t_1, c_p) * Math.pow((1.0 - t_1), c_n));
}

Python

def code(c_p, c_n, t, s):
	t_1 = 1.0 / (1.0 + math.exp(-t))
	t_2 = 1.0 / (1.0 + math.exp(-s))
	return (math.pow(t_2, c_p) * math.pow((1.0 - t_2), c_n)) / (math.pow(t_1, c_p) * math.pow((1.0 - t_1), c_n))

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(1.0 / Float64(1.0 + exp(Float64(-t))))
	t_2 = Float64(1.0 / Float64(1.0 + exp(Float64(-s))))
	return Float64(Float64((t_2 ^ c_p) * (Float64(1.0 - t_2) ^ c_n)) / Float64((t_1 ^ c_p) * (Float64(1.0 - t_1) ^ c_n)))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	t_1 = 1.0 / (1.0 + exp(-t));
	t_2 = 1.0 / (1.0 + exp(-s));
	tmp = ((t_2 ^ c_p) * ((1.0 - t_2) ^ c_n)) / ((t_1 ^ c_p) * ((1.0 - t_1) ^ c_n));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(1.0 / N[(1.0 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[t$95$2, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$2), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$1, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$1), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{1 + e^{-t}}\\ t_2 := \frac{1}{1 + e^{-s}}\\ \frac{{t\_2}^{c\_p} \cdot {\left(1 - t\_2\right)}^{c\_n}}{{t\_1}^{c\_p} \cdot {\left(1 - t\_1\right)}^{c\_n}} \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ 1.0 (exp (- t))))) (t_2 (/ 1.0 (+ 1.0 (exp (- s))))))
   (/
    (* (pow t_2 c_p) (pow (- 1.0 t_2) c_n))
    (* (pow t_1 c_p) (pow (- 1.0 t_1) c_n)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 / (1.0 + exp(-t));
	double t_2 = 1.0 / (1.0 + exp(-s));
	return (pow(t_2, c_p) * pow((1.0 - t_2), c_n)) / (pow(t_1, c_p) * pow((1.0 - t_1), c_n));
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 1.0d0 / (1.0d0 + exp(-t))
    t_2 = 1.0d0 / (1.0d0 + exp(-s))
    code = ((t_2 ** c_p) * ((1.0d0 - t_2) ** c_n)) / ((t_1 ** c_p) * ((1.0d0 - t_1) ** c_n))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 / (1.0 + Math.exp(-t));
	double t_2 = 1.0 / (1.0 + Math.exp(-s));
	return (Math.pow(t_2, c_p) * Math.pow((1.0 - t_2), c_n)) / (Math.pow(t_1, c_p) * Math.pow((1.0 - t_1), c_n));
}

Python

def code(c_p, c_n, t, s):
	t_1 = 1.0 / (1.0 + math.exp(-t))
	t_2 = 1.0 / (1.0 + math.exp(-s))
	return (math.pow(t_2, c_p) * math.pow((1.0 - t_2), c_n)) / (math.pow(t_1, c_p) * math.pow((1.0 - t_1), c_n))

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(1.0 / Float64(1.0 + exp(Float64(-t))))
	t_2 = Float64(1.0 / Float64(1.0 + exp(Float64(-s))))
	return Float64(Float64((t_2 ^ c_p) * (Float64(1.0 - t_2) ^ c_n)) / Float64((t_1 ^ c_p) * (Float64(1.0 - t_1) ^ c_n)))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	t_1 = 1.0 / (1.0 + exp(-t));
	t_2 = 1.0 / (1.0 + exp(-s));
	tmp = ((t_2 ^ c_p) * ((1.0 - t_2) ^ c_n)) / ((t_1 ^ c_p) * ((1.0 - t_1) ^ c_n));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(1.0 / N[(1.0 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[t$95$2, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$2), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$1, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$1), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 98.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ e^{0.5 \cdot \left(c\_n \cdot t\right) + s \cdot \left(c\_n \cdot -0.5 + -0.125 \cdot \left(c\_n \cdot s\right)\right)} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (exp (+ (* 0.5 (* c_n t)) (* s (+ (* c_n -0.5) (* -0.125 (* c_n s)))))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp(((0.5 * (c_n * t)) + (s * ((c_n * -0.5) + (-0.125 * (c_n * s))))));
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = exp(((0.5d0 * (c_n * t)) + (s * ((c_n * (-0.5d0)) + ((-0.125d0) * (c_n * s))))))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.exp(((0.5 * (c_n * t)) + (s * ((c_n * -0.5) + (-0.125 * (c_n * s))))));
}

Python

def code(c_p, c_n, t, s):
	return math.exp(((0.5 * (c_n * t)) + (s * ((c_n * -0.5) + (-0.125 * (c_n * s))))))

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(Float64(0.5 * Float64(c_n * t)) + Float64(s * Float64(Float64(c_n * -0.5) + Float64(-0.125 * Float64(c_n * s))))))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = exp(((0.5 * (c_n * t)) + (s * ((c_n * -0.5) + (-0.125 * (c_n * s))))));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[(N[(0.5 * N[(c$95$n * t), $MachinePrecision]), $MachinePrecision] + N[(s * N[(N[(c$95$n * -0.5), $MachinePrecision] + N[(-0.125 * N[(c$95$n * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 90.3%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
2. Step-by-step derivation

   1. associate-/l/ 90.3%

      \[\leadsto \color{blue}{\frac{\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}}} \]

3. Simplified 90.3%

   \[\leadsto \color{blue}{\frac{\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 + \frac{1}{-1 - e^{-s}}\right)}^{c\_n}}{{\left(1 + \frac{1}{-1 - e^{-t}}\right)}^{c\_n}}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}}} \]
4. Add Preprocessing

5. Taylor expanded in c_p around 0 94.5%

   \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}}} \]
6. Step-by-step derivation

   1. pow-to-exp 94.4%

      \[\leadsto \frac{\color{blue}{e^{\log \left(1 - \frac{1}{1 + e^{-s}}\right) \cdot c\_n}}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]

   2. pow-to-exp 94.5%

      \[\leadsto \frac{e^{\log \left(1 - \frac{1}{1 + e^{-s}}\right) \cdot c\_n}}{\color{blue}{e^{\log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c\_n}}} \]

   3. div-exp 96.3%

      \[\leadsto \color{blue}{e^{\log \left(1 - \frac{1}{1 + e^{-s}}\right) \cdot c\_n - \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c\_n}} \]

   4. sub-neg 96.3%

      \[\leadsto e^{\log \color{blue}{\left(1 + \left(-\frac{1}{1 + e^{-s}}\right)\right)} \cdot c\_n - \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c\_n} \]

   5. log1p-define 96.3%

      \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(-\frac{1}{1 + e^{-s}}\right)} \cdot c\_n - \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c\_n} \]

   6. sub-neg 96.3%

      \[\leadsto e^{\mathsf{log1p}\left(-\frac{1}{1 + e^{-s}}\right) \cdot c\_n - \log \color{blue}{\left(1 + \left(-\frac{1}{1 + e^{-t}}\right)\right)} \cdot c\_n} \]

   7. log1p-define 96.3%

      \[\leadsto e^{\mathsf{log1p}\left(-\frac{1}{1 + e^{-s}}\right) \cdot c\_n - \color{blue}{\mathsf{log1p}\left(-\frac{1}{1 + e^{-t}}\right)} \cdot c\_n} \]

7. Applied egg-rr 96.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{1}{1 + e^{-s}}\right) \cdot c\_n - \mathsf{log1p}\left(-\frac{1}{1 + e^{-t}}\right) \cdot c\_n}} \]

8. Taylor expanded in t around 0 96.4%

   \[\leadsto e^{\color{blue}{\left(0.5 \cdot \left(c\_n \cdot t\right) + c\_n \cdot \log \left(1 - \frac{1}{1 + e^{-s}}\right)\right) - c\_n \cdot \log 0.5}} \]
9. Step-by-step derivation

   1. associate--l+ 96.5%

      \[\leadsto e^{\color{blue}{0.5 \cdot \left(c\_n \cdot t\right) + \left(c\_n \cdot \log \left(1 - \frac{1}{1 + e^{-s}}\right) - c\_n \cdot \log 0.5\right)}} \]

   2. distribute-lft-out-- 96.5%

      \[\leadsto e^{0.5 \cdot \left(c\_n \cdot t\right) + \color{blue}{c\_n \cdot \left(\log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log 0.5\right)}} \]

   3. +-commutative 96.5%

      \[\leadsto e^{\color{blue}{c\_n \cdot \left(\log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log 0.5\right) + 0.5 \cdot \left(c\_n \cdot t\right)}} \]

   4. fma-define 96.5%

      \[\leadsto e^{\color{blue}{\mathsf{fma}\left(c\_n, \log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log 0.5, 0.5 \cdot \left(c\_n \cdot t\right)\right)}} \]

   5. sub-neg 96.5%

      \[\leadsto e^{\mathsf{fma}\left(c\_n, \log \color{blue}{\left(1 + \left(-\frac{1}{1 + e^{-s}}\right)\right)} - \log 0.5, 0.5 \cdot \left(c\_n \cdot t\right)\right)} \]

   6. log1p-define 96.5%

      \[\leadsto e^{\mathsf{fma}\left(c\_n, \color{blue}{\mathsf{log1p}\left(-\frac{1}{1 + e^{-s}}\right)} - \log 0.5, 0.5 \cdot \left(c\_n \cdot t\right)\right)} \]

   7. distribute-neg-frac 96.5%

      \[\leadsto e^{\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\color{blue}{\frac{-1}{1 + e^{-s}}}\right) - \log 0.5, 0.5 \cdot \left(c\_n \cdot t\right)\right)} \]

   8. metadata-eval 96.5%

      \[\leadsto e^{\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{\color{blue}{-1}}{1 + e^{-s}}\right) - \log 0.5, 0.5 \cdot \left(c\_n \cdot t\right)\right)} \]

   9. *-commutative 96.5%

      \[\leadsto e^{\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right) - \log 0.5, \color{blue}{\left(c\_n \cdot t\right) \cdot 0.5}\right)} \]

   10. associate-*l* 96.5%

       \[\leadsto e^{\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right) - \log 0.5, \color{blue}{c\_n \cdot \left(t \cdot 0.5\right)}\right)} \]

   11. *-commutative 96.5%

       \[\leadsto e^{\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right) - \log 0.5, c\_n \cdot \color{blue}{\left(0.5 \cdot t\right)}\right)} \]

10. Simplified 96.5%

    \[\leadsto e^{\color{blue}{\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right) - \log 0.5, c\_n \cdot \left(0.5 \cdot t\right)\right)}} \]

11. Taylor expanded in s around 0 99.5%

    \[\leadsto e^{\color{blue}{0.5 \cdot \left(c\_n \cdot t\right) + s \cdot \left(-0.5 \cdot c\_n + -0.125 \cdot \left(c\_n \cdot s\right)\right)}} \]

12. Final simplification 99.5%

    \[\leadsto e^{0.5 \cdot \left(c\_n \cdot t\right) + s \cdot \left(c\_n \cdot -0.5 + -0.125 \cdot \left(c\_n \cdot s\right)\right)} \]
13. Add Preprocessing

Alternative 2: 96.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ e^{0.5 \cdot \left(c\_n \cdot t\right) + -0.5 \cdot \left(c\_n \cdot s\right)} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (exp (+ (* 0.5 (* c_n t)) (* -0.5 (* c_n s)))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp(((0.5 * (c_n * t)) + (-0.5 * (c_n * s))));
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = exp(((0.5d0 * (c_n * t)) + ((-0.5d0) * (c_n * s))))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.exp(((0.5 * (c_n * t)) + (-0.5 * (c_n * s))));
}

Python

def code(c_p, c_n, t, s):
	return math.exp(((0.5 * (c_n * t)) + (-0.5 * (c_n * s))))

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(Float64(0.5 * Float64(c_n * t)) + Float64(-0.5 * Float64(c_n * s))))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = exp(((0.5 * (c_n * t)) + (-0.5 * (c_n * s))));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[(N[(0.5 * N[(c$95$n * t), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c$95$n * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 90.3%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
2. Step-by-step derivation

   1. associate-/l/ 90.3%

      \[\leadsto \color{blue}{\frac{\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}}} \]

3. Simplified 90.3%

   \[\leadsto \color{blue}{\frac{\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 + \frac{1}{-1 - e^{-s}}\right)}^{c\_n}}{{\left(1 + \frac{1}{-1 - e^{-t}}\right)}^{c\_n}}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}}} \]
4. Add Preprocessing

5. Taylor expanded in c_p around 0 94.5%

   \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}}} \]
6. Step-by-step derivation

   1. pow-to-exp 94.4%

      \[\leadsto \frac{\color{blue}{e^{\log \left(1 - \frac{1}{1 + e^{-s}}\right) \cdot c\_n}}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]

   2. pow-to-exp 94.5%

      \[\leadsto \frac{e^{\log \left(1 - \frac{1}{1 + e^{-s}}\right) \cdot c\_n}}{\color{blue}{e^{\log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c\_n}}} \]

   3. div-exp 96.3%

      \[\leadsto \color{blue}{e^{\log \left(1 - \frac{1}{1 + e^{-s}}\right) \cdot c\_n - \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c\_n}} \]

   4. sub-neg 96.3%

      \[\leadsto e^{\log \color{blue}{\left(1 + \left(-\frac{1}{1 + e^{-s}}\right)\right)} \cdot c\_n - \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c\_n} \]

   5. log1p-define 96.3%

      \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(-\frac{1}{1 + e^{-s}}\right)} \cdot c\_n - \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c\_n} \]

   6. sub-neg 96.3%

      \[\leadsto e^{\mathsf{log1p}\left(-\frac{1}{1 + e^{-s}}\right) \cdot c\_n - \log \color{blue}{\left(1 + \left(-\frac{1}{1 + e^{-t}}\right)\right)} \cdot c\_n} \]

   7. log1p-define 96.3%

      \[\leadsto e^{\mathsf{log1p}\left(-\frac{1}{1 + e^{-s}}\right) \cdot c\_n - \color{blue}{\mathsf{log1p}\left(-\frac{1}{1 + e^{-t}}\right)} \cdot c\_n} \]

7. Applied egg-rr 96.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{1}{1 + e^{-s}}\right) \cdot c\_n - \mathsf{log1p}\left(-\frac{1}{1 + e^{-t}}\right) \cdot c\_n}} \]

8. Taylor expanded in t around 0 96.4%

   \[\leadsto e^{\color{blue}{\left(0.5 \cdot \left(c\_n \cdot t\right) + c\_n \cdot \log \left(1 - \frac{1}{1 + e^{-s}}\right)\right) - c\_n \cdot \log 0.5}} \]
9. Step-by-step derivation

   1. associate--l+ 96.5%

      \[\leadsto e^{\color{blue}{0.5 \cdot \left(c\_n \cdot t\right) + \left(c\_n \cdot \log \left(1 - \frac{1}{1 + e^{-s}}\right) - c\_n \cdot \log 0.5\right)}} \]

   2. distribute-lft-out-- 96.5%

      \[\leadsto e^{0.5 \cdot \left(c\_n \cdot t\right) + \color{blue}{c\_n \cdot \left(\log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log 0.5\right)}} \]

   3. +-commutative 96.5%

      \[\leadsto e^{\color{blue}{c\_n \cdot \left(\log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log 0.5\right) + 0.5 \cdot \left(c\_n \cdot t\right)}} \]

   4. fma-define 96.5%

      \[\leadsto e^{\color{blue}{\mathsf{fma}\left(c\_n, \log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log 0.5, 0.5 \cdot \left(c\_n \cdot t\right)\right)}} \]

   5. sub-neg 96.5%

      \[\leadsto e^{\mathsf{fma}\left(c\_n, \log \color{blue}{\left(1 + \left(-\frac{1}{1 + e^{-s}}\right)\right)} - \log 0.5, 0.5 \cdot \left(c\_n \cdot t\right)\right)} \]

   6. log1p-define 96.5%

      \[\leadsto e^{\mathsf{fma}\left(c\_n, \color{blue}{\mathsf{log1p}\left(-\frac{1}{1 + e^{-s}}\right)} - \log 0.5, 0.5 \cdot \left(c\_n \cdot t\right)\right)} \]

   7. distribute-neg-frac 96.5%

      \[\leadsto e^{\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\color{blue}{\frac{-1}{1 + e^{-s}}}\right) - \log 0.5, 0.5 \cdot \left(c\_n \cdot t\right)\right)} \]

   8. metadata-eval 96.5%

      \[\leadsto e^{\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{\color{blue}{-1}}{1 + e^{-s}}\right) - \log 0.5, 0.5 \cdot \left(c\_n \cdot t\right)\right)} \]

   9. *-commutative 96.5%

      \[\leadsto e^{\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right) - \log 0.5, \color{blue}{\left(c\_n \cdot t\right) \cdot 0.5}\right)} \]

   10. associate-*l* 96.5%

       \[\leadsto e^{\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right) - \log 0.5, \color{blue}{c\_n \cdot \left(t \cdot 0.5\right)}\right)} \]

   11. *-commutative 96.5%

       \[\leadsto e^{\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right) - \log 0.5, c\_n \cdot \color{blue}{\left(0.5 \cdot t\right)}\right)} \]

10. Simplified 96.5%

    \[\leadsto e^{\color{blue}{\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right) - \log 0.5, c\_n \cdot \left(0.5 \cdot t\right)\right)}} \]

11. Taylor expanded in s around 0 96.8%

    \[\leadsto e^{\color{blue}{-0.5 \cdot \left(c\_n \cdot s\right) + 0.5 \cdot \left(c\_n \cdot t\right)}} \]

12. Final simplification 96.8%

    \[\leadsto e^{0.5 \cdot \left(c\_n \cdot t\right) + -0.5 \cdot \left(c\_n \cdot s\right)} \]
13. Add Preprocessing

Alternative 3: 95.3% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ e^{c\_n \cdot \left(0.5 \cdot t\right)} \end{array} \]

FPCore

(FPCore (c_p c_n t s) :precision binary64 (exp (* c_n (* 0.5 t))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp((c_n * (0.5 * t)));
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = exp((c_n * (0.5d0 * t)))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.exp((c_n * (0.5 * t)));
}

Python

def code(c_p, c_n, t, s):
	return math.exp((c_n * (0.5 * t)))

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(c_n * Float64(0.5 * t)))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = exp((c_n * (0.5 * t)));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[(c$95$n * N[(0.5 * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 90.3%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
2. Step-by-step derivation

   1. associate-/l/ 90.3%

      \[\leadsto \color{blue}{\frac{\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}}} \]

3. Simplified 90.3%

   \[\leadsto \color{blue}{\frac{\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 + \frac{1}{-1 - e^{-s}}\right)}^{c\_n}}{{\left(1 + \frac{1}{-1 - e^{-t}}\right)}^{c\_n}}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}}} \]
4. Add Preprocessing

5. Taylor expanded in c_p around 0 94.5%

   \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}}} \]
6. Step-by-step derivation

   1. pow-to-exp 94.4%

      \[\leadsto \frac{\color{blue}{e^{\log \left(1 - \frac{1}{1 + e^{-s}}\right) \cdot c\_n}}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]

   2. pow-to-exp 94.5%

      \[\leadsto \frac{e^{\log \left(1 - \frac{1}{1 + e^{-s}}\right) \cdot c\_n}}{\color{blue}{e^{\log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c\_n}}} \]

   3. div-exp 96.3%

      \[\leadsto \color{blue}{e^{\log \left(1 - \frac{1}{1 + e^{-s}}\right) \cdot c\_n - \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c\_n}} \]

   4. sub-neg 96.3%

      \[\leadsto e^{\log \color{blue}{\left(1 + \left(-\frac{1}{1 + e^{-s}}\right)\right)} \cdot c\_n - \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c\_n} \]

   5. log1p-define 96.3%

      \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(-\frac{1}{1 + e^{-s}}\right)} \cdot c\_n - \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c\_n} \]

   6. sub-neg 96.3%

      \[\leadsto e^{\mathsf{log1p}\left(-\frac{1}{1 + e^{-s}}\right) \cdot c\_n - \log \color{blue}{\left(1 + \left(-\frac{1}{1 + e^{-t}}\right)\right)} \cdot c\_n} \]

   7. log1p-define 96.3%

      \[\leadsto e^{\mathsf{log1p}\left(-\frac{1}{1 + e^{-s}}\right) \cdot c\_n - \color{blue}{\mathsf{log1p}\left(-\frac{1}{1 + e^{-t}}\right)} \cdot c\_n} \]

7. Applied egg-rr 96.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{1}{1 + e^{-s}}\right) \cdot c\_n - \mathsf{log1p}\left(-\frac{1}{1 + e^{-t}}\right) \cdot c\_n}} \]

8. Taylor expanded in t around 0 96.4%

   \[\leadsto e^{\color{blue}{\left(0.5 \cdot \left(c\_n \cdot t\right) + c\_n \cdot \log \left(1 - \frac{1}{1 + e^{-s}}\right)\right) - c\_n \cdot \log 0.5}} \]
9. Step-by-step derivation

   1. associate--l+ 96.5%

      \[\leadsto e^{\color{blue}{0.5 \cdot \left(c\_n \cdot t\right) + \left(c\_n \cdot \log \left(1 - \frac{1}{1 + e^{-s}}\right) - c\_n \cdot \log 0.5\right)}} \]

   2. distribute-lft-out-- 96.5%

      \[\leadsto e^{0.5 \cdot \left(c\_n \cdot t\right) + \color{blue}{c\_n \cdot \left(\log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log 0.5\right)}} \]

   3. +-commutative 96.5%

      \[\leadsto e^{\color{blue}{c\_n \cdot \left(\log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log 0.5\right) + 0.5 \cdot \left(c\_n \cdot t\right)}} \]

   4. fma-define 96.5%

      \[\leadsto e^{\color{blue}{\mathsf{fma}\left(c\_n, \log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log 0.5, 0.5 \cdot \left(c\_n \cdot t\right)\right)}} \]

   5. sub-neg 96.5%

      \[\leadsto e^{\mathsf{fma}\left(c\_n, \log \color{blue}{\left(1 + \left(-\frac{1}{1 + e^{-s}}\right)\right)} - \log 0.5, 0.5 \cdot \left(c\_n \cdot t\right)\right)} \]

   6. log1p-define 96.5%

      \[\leadsto e^{\mathsf{fma}\left(c\_n, \color{blue}{\mathsf{log1p}\left(-\frac{1}{1 + e^{-s}}\right)} - \log 0.5, 0.5 \cdot \left(c\_n \cdot t\right)\right)} \]

   7. distribute-neg-frac 96.5%

      \[\leadsto e^{\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\color{blue}{\frac{-1}{1 + e^{-s}}}\right) - \log 0.5, 0.5 \cdot \left(c\_n \cdot t\right)\right)} \]

   8. metadata-eval 96.5%

      \[\leadsto e^{\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{\color{blue}{-1}}{1 + e^{-s}}\right) - \log 0.5, 0.5 \cdot \left(c\_n \cdot t\right)\right)} \]

   9. *-commutative 96.5%

      \[\leadsto e^{\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right) - \log 0.5, \color{blue}{\left(c\_n \cdot t\right) \cdot 0.5}\right)} \]

   10. associate-*l* 96.5%

       \[\leadsto e^{\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right) - \log 0.5, \color{blue}{c\_n \cdot \left(t \cdot 0.5\right)}\right)} \]

   11. *-commutative 96.5%

       \[\leadsto e^{\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right) - \log 0.5, c\_n \cdot \color{blue}{\left(0.5 \cdot t\right)}\right)} \]

10. Simplified 96.5%

    \[\leadsto e^{\color{blue}{\mathsf{fma}\left(c\_n, \mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right) - \log 0.5, c\_n \cdot \left(0.5 \cdot t\right)\right)}} \]

11. Taylor expanded in s around 0 95.7%

    \[\leadsto e^{\color{blue}{0.5 \cdot \left(c\_n \cdot t\right)}} \]
12. Step-by-step derivation

    1. associate-*r* 95.7%

       \[\leadsto e^{\color{blue}{\left(0.5 \cdot c\_n\right) \cdot t}} \]

    2. *-commutative 95.7%

       \[\leadsto e^{\color{blue}{\left(c\_n \cdot 0.5\right)} \cdot t} \]

    3. associate-*r* 95.7%

       \[\leadsto e^{\color{blue}{c\_n \cdot \left(0.5 \cdot t\right)}} \]

13. Simplified 95.7%

    \[\leadsto e^{\color{blue}{c\_n \cdot \left(0.5 \cdot t\right)}} \]
14. Add Preprocessing

Alternative 4: 94.0% accurate, 835.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (c_p c_n t s) :precision binary64 1.0)

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = 1.0d0
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 1.0;
}

Python

def code(c_p, c_n, t, s):
	return 1.0

Julia

function code(c_p, c_n, t, s)
	return 1.0
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := 1.0

Derivation

1. Initial program 90.3%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
2. Step-by-step derivation

   1. associate-/l/ 90.3%

      \[\leadsto \color{blue}{\frac{\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}}} \]

3. Simplified 90.3%

   \[\leadsto \color{blue}{\frac{\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 + \frac{1}{-1 - e^{-s}}\right)}^{c\_n}}{{\left(1 + \frac{1}{-1 - e^{-t}}\right)}^{c\_n}}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}}} \]
4. Add Preprocessing

5. Taylor expanded in c_n around 0 91.7%

   \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}}} \]

6. Taylor expanded in c_p around 0 86.6%

   \[\leadsto \frac{\color{blue}{1}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}} \]

7. Taylor expanded in c_p around 0 94.4%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Developer Target 1: 96.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1 + e^{-t}}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(\frac{1 + e^{t}}{1 + e^{s}}\right)}^{c\_n} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (*
  (pow (/ (+ 1.0 (exp (- t))) (+ 1.0 (exp (- s)))) c_p)
  (pow (/ (+ 1.0 (exp t)) (+ 1.0 (exp s))) c_n)))

C

double code(double c_p, double c_n, double t, double s) {
	return pow(((1.0 + exp(-t)) / (1.0 + exp(-s))), c_p) * pow(((1.0 + exp(t)) / (1.0 + exp(s))), c_n);
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = (((1.0d0 + exp(-t)) / (1.0d0 + exp(-s))) ** c_p) * (((1.0d0 + exp(t)) / (1.0d0 + exp(s))) ** c_n)
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.pow(((1.0 + Math.exp(-t)) / (1.0 + Math.exp(-s))), c_p) * Math.pow(((1.0 + Math.exp(t)) / (1.0 + Math.exp(s))), c_n);
}

Python

def code(c_p, c_n, t, s):
	return math.pow(((1.0 + math.exp(-t)) / (1.0 + math.exp(-s))), c_p) * math.pow(((1.0 + math.exp(t)) / (1.0 + math.exp(s))), c_n)

Julia

function code(c_p, c_n, t, s)
	return Float64((Float64(Float64(1.0 + exp(Float64(-t))) / Float64(1.0 + exp(Float64(-s)))) ^ c_p) * (Float64(Float64(1.0 + exp(t)) / Float64(1.0 + exp(s))) ^ c_n))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = (((1.0 + exp(-t)) / (1.0 + exp(-s))) ^ c_p) * (((1.0 + exp(t)) / (1.0 + exp(s))) ^ c_n);
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(N[Power[N[(N[(1.0 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision] * N[Power[N[(N[(1.0 + N[Exp[t], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Exp[s], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024180 
(FPCore (c_p c_n t s)
  :name "Harley's example"
  :precision binary64
  :pre (and (< 0.0 c_p) (< 0.0 c_n))

  :alt
  (! :herbie-platform default (* (pow (/ (+ 1 (exp (- t))) (+ 1 (exp (- s)))) c_p) (pow (/ (+ 1 (exp t)) (+ 1 (exp s))) c_n)))

  (/ (* (pow (/ 1.0 (+ 1.0 (exp (- s)))) c_p) (pow (- 1.0 (/ 1.0 (+ 1.0 (exp (- s))))) c_n)) (* (pow (/ 1.0 (+ 1.0 (exp (- t)))) c_p) (pow (- 1.0 (/ 1.0 (+ 1.0 (exp (- t))))) c_n))))