Harley's example

Percentage Accurate: 90.5% → 96.3%

Time: 2.3min

Alternatives: 5

Speedup: 835.0×

• could not determine a ground truth

  1. c_p = 2.00000000000000007e154
  2. c_n = 2
  3. t = 0.0
  4. s = 0.0

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: 90.5% 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: 96.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double c_p, double c_n, double t, double s) {
	return exp((c_n * (log((1.0 + (1.0 / (1.0 + exp(s))))) - log((1.0 + (1.0 / (1.0 + exp(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 * (log((1.0d0 + (1.0d0 / (1.0d0 + exp(s))))) - log((1.0d0 + (1.0d0 / (1.0d0 + exp(t))))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

6. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{\left(\left(c\_p \cdot \left(-\mathsf{log1p}\left(e^{s}\right)\right) + c\_n \cdot \mathsf{log1p}\left(\frac{1}{1 + e^{s}}\right)\right) - c\_n \cdot \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{t}\right)\right)}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-define 98.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\left(c\_p \cdot \left(-\mathsf{log1p}\left(e^{s}\right)\right) + c\_n \cdot \mathsf{log1p}\left(\frac{1}{1 + e^{s}}\right)\right) - c\_n \cdot \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{t}\right)\right)}\right)\right)} \]

   2. associate--l+ 98.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\left(c\_p \cdot \left(-\mathsf{log1p}\left(e^{s}\right)\right) + \left(c\_n \cdot \mathsf{log1p}\left(\frac{1}{1 + e^{s}}\right) - c\_n \cdot \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right)\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{t}\right)\right)}\right)\right) \]

   3. distribute-lft-out-- 98.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(c\_p \cdot \left(-\mathsf{log1p}\left(e^{s}\right)\right) + \color{blue}{c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{1 + e^{s}}\right) - \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right)}\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{t}\right)\right)}\right)\right) \]

8. Simplified 98.2%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(c\_p \cdot \left(-\mathsf{log1p}\left(e^{s}\right)\right) + c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{1 + e^{s}}\right) - \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right)\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{t}\right)\right)}\right)\right)} \]

9. Taylor expanded in c_p around 0 98.2%

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

Alternative 2: 95.5% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ e^{t \cdot \left(c\_n \cdot 0.16666666666666666\right)} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (exp (* t (* c_n 0.16666666666666666))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp((t * (c_n * 0.16666666666666666)));
}

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((t * (c_n * 0.16666666666666666d0)))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.exp((t * (c_n * 0.16666666666666666)));
}

Python

def code(c_p, c_n, t, s):
	return math.exp((t * (c_n * 0.16666666666666666)))

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(t * Float64(c_n * 0.16666666666666666)))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = exp((t * (c_n * 0.16666666666666666)));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[(t * N[(c$95$n * 0.16666666666666666), $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

6. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{\left(\left(c\_p \cdot \left(-\mathsf{log1p}\left(e^{s}\right)\right) + c\_n \cdot \mathsf{log1p}\left(\frac{1}{1 + e^{s}}\right)\right) - c\_n \cdot \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{t}\right)\right)}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-define 98.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\left(c\_p \cdot \left(-\mathsf{log1p}\left(e^{s}\right)\right) + c\_n \cdot \mathsf{log1p}\left(\frac{1}{1 + e^{s}}\right)\right) - c\_n \cdot \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{t}\right)\right)}\right)\right)} \]

   2. associate--l+ 98.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\left(c\_p \cdot \left(-\mathsf{log1p}\left(e^{s}\right)\right) + \left(c\_n \cdot \mathsf{log1p}\left(\frac{1}{1 + e^{s}}\right) - c\_n \cdot \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right)\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{t}\right)\right)}\right)\right) \]

   3. distribute-lft-out-- 98.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(c\_p \cdot \left(-\mathsf{log1p}\left(e^{s}\right)\right) + \color{blue}{c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{1 + e^{s}}\right) - \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right)}\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{t}\right)\right)}\right)\right) \]

8. Simplified 98.2%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(c\_p \cdot \left(-\mathsf{log1p}\left(e^{s}\right)\right) + c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{1 + e^{s}}\right) - \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right)\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{t}\right)\right)}\right)\right)} \]

9. Taylor expanded in c_p around 0 98.2%

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

10. Taylor expanded in s around 0 97.9%

    \[\leadsto e^{\color{blue}{c\_n \cdot \left(\log 1.5 - \log \left(1 + \frac{1}{1 + e^{t}}\right)\right)}} \]
11. Step-by-step derivation

    1. log1p-define 97.9%

       \[\leadsto e^{c\_n \cdot \left(\log 1.5 - \color{blue}{\mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)}\right)} \]

12. Simplified 97.9%

    \[\leadsto e^{\color{blue}{c\_n \cdot \left(\log 1.5 - \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right)}} \]

13. Taylor expanded in t around 0 97.9%

    \[\leadsto e^{\color{blue}{0.16666666666666666 \cdot \left(c\_n \cdot t\right)}} \]
14. Step-by-step derivation

    1. *-commutative 97.9%

       \[\leadsto e^{\color{blue}{\left(c\_n \cdot t\right) \cdot 0.16666666666666666}} \]

    2. *-commutative 97.9%

       \[\leadsto e^{\color{blue}{\left(t \cdot c\_n\right)} \cdot 0.16666666666666666} \]

    3. associate-*r* 97.9%

       \[\leadsto e^{\color{blue}{t \cdot \left(c\_n \cdot 0.16666666666666666\right)}} \]

15. Simplified 97.9%

    \[\leadsto e^{\color{blue}{t \cdot \left(c\_n \cdot 0.16666666666666666\right)}} \]
16. Add Preprocessing

Alternative 3: 94.1% accurate, 64.2× speedup

Math

\[\begin{array}{l} \\ 1 + \left(0.5 \cdot \left(c\_n \cdot t\right) + c\_n \cdot \left(s \cdot -0.5\right)\right) \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (+ 1.0 (+ (* 0.5 (* c_n t)) (* c_n (* s -0.5)))))

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0 + ((0.5 * (c_n * t)) + (c_n * (s * -0.5)));
}

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 + ((0.5d0 * (c_n * t)) + (c_n * (s * (-0.5d0))))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 1.0 + ((0.5 * (c_n * t)) + (c_n * (s * -0.5)));
}

Python

def code(c_p, c_n, t, s):
	return 1.0 + ((0.5 * (c_n * t)) + (c_n * (s * -0.5)))

Julia

function code(c_p, c_n, t, s)
	return Float64(1.0 + Float64(Float64(0.5 * Float64(c_n * t)) + Float64(c_n * Float64(s * -0.5))))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0 + ((0.5 * (c_n * t)) + (c_n * (s * -0.5)));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(1.0 + N[(N[(0.5 * N[(c$95$n * t), $MachinePrecision]), $MachinePrecision] + N[(c$95$n * N[(s * -0.5), $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_n around 0 93.3%

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

   1. fma-define 93.3%

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

7. Simplified 93.3%

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

8. Taylor expanded in t around 0 94.5%

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

9. Taylor expanded in c_p around 0 96.6%

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

10. Taylor expanded in s around 0 96.5%

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

    1. *-commutative 96.5%

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

12. Simplified 96.5%

    \[\leadsto 1 + \left(0.5 \cdot \left(c\_n \cdot t\right) + c\_n \cdot \color{blue}{\left(s \cdot -0.5\right)}\right) \]
13. Add Preprocessing

Alternative 4: 94.1% accurate, 119.3× speedup

Math

\[\begin{array}{l} \\ 1 + t \cdot \left(c\_n \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (c_p c_n t s) :precision binary64 (+ 1.0 (* t (* c_n 0.5))))

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0 + (t * (c_n * 0.5));
}

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 + (t * (c_n * 0.5d0))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 1.0 + (t * (c_n * 0.5));
}

Python

def code(c_p, c_n, t, s):
	return 1.0 + (t * (c_n * 0.5))

Julia

function code(c_p, c_n, t, s)
	return Float64(1.0 + Float64(t * Float64(c_n * 0.5)))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0 + (t * (c_n * 0.5));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(1.0 + N[(t * N[(c$95$n * 0.5), $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 94.1%

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

6. Taylor expanded in s around 0 93.9%

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

7. Taylor expanded in t around 0 96.4%

   \[\leadsto \color{blue}{1 + 0.5 \cdot \left(c\_n \cdot t\right)} \]
8. Step-by-step derivation

   1. associate-*r* 96.4%

      \[\leadsto 1 + \color{blue}{\left(0.5 \cdot c\_n\right) \cdot t} \]

   2. *-commutative 96.4%

      \[\leadsto 1 + \color{blue}{\left(c\_n \cdot 0.5\right)} \cdot t \]

9. Simplified 96.4%

   \[\leadsto \color{blue}{1 + \left(c\_n \cdot 0.5\right) \cdot t} \]

10. Final simplification 96.4%

    \[\leadsto 1 + t \cdot \left(c\_n \cdot 0.5\right) \]
11. Add Preprocessing

Alternative 5: 94.1% 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 94.3%

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

6. Taylor expanded in c_p around 0 96.3%

   \[\leadsto \color{blue}{1} \]
7. 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 2024163 
(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))))