Harley's example

Percentage Accurate: 90.9% → 96.7%

Time: 1.4min

Alternatives: 6

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: 90.9% 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: 96.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{-t}\\ t_2 := e^{-s}\\ \mathbf{if}\;c\_p \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{{\left(1 + \frac{1}{-1 - t\_2}\right)}^{c\_n}}{{\left(1 + \frac{1}{-1 - t\_1}\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;e^{c\_p \cdot \left(\mathsf{log1p}\left(t\_1\right) - \mathsf{log1p}\left(t\_2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (exp (- t))) (t_2 (exp (- s))))
   (if (<= c_p 2e-27)
     (/
      (pow (+ 1.0 (/ 1.0 (- -1.0 t_2))) c_n)
      (pow (+ 1.0 (/ 1.0 (- -1.0 t_1))) c_n))
     (exp (* c_p (- (log1p t_1) (log1p t_2)))))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = exp(-t);
	double t_2 = exp(-s);
	double tmp;
	if (c_p <= 2e-27) {
		tmp = pow((1.0 + (1.0 / (-1.0 - t_2))), c_n) / pow((1.0 + (1.0 / (-1.0 - t_1))), c_n);
	} else {
		tmp = exp((c_p * (log1p(t_1) - log1p(t_2))));
	}
	return tmp;
}

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = Math.exp(-t);
	double t_2 = Math.exp(-s);
	double tmp;
	if (c_p <= 2e-27) {
		tmp = Math.pow((1.0 + (1.0 / (-1.0 - t_2))), c_n) / Math.pow((1.0 + (1.0 / (-1.0 - t_1))), c_n);
	} else {
		tmp = Math.exp((c_p * (Math.log1p(t_1) - Math.log1p(t_2))));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	t_1 = math.exp(-t)
	t_2 = math.exp(-s)
	tmp = 0
	if c_p <= 2e-27:
		tmp = math.pow((1.0 + (1.0 / (-1.0 - t_2))), c_n) / math.pow((1.0 + (1.0 / (-1.0 - t_1))), c_n)
	else:
		tmp = math.exp((c_p * (math.log1p(t_1) - math.log1p(t_2))))
	return tmp

Julia

function code(c_p, c_n, t, s)
	t_1 = exp(Float64(-t))
	t_2 = exp(Float64(-s))
	tmp = 0.0
	if (c_p <= 2e-27)
		tmp = Float64((Float64(1.0 + Float64(1.0 / Float64(-1.0 - t_2))) ^ c_n) / (Float64(1.0 + Float64(1.0 / Float64(-1.0 - t_1))) ^ c_n));
	else
		tmp = exp(Float64(c_p * Float64(log1p(t_1) - log1p(t_2))));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Exp[(-t)], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-s)], $MachinePrecision]}, If[LessEqual[c$95$p, 2e-27], N[(N[Power[N[(1.0 + N[(1.0 / N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision] / N[Power[N[(1.0 + N[(1.0 / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision], N[Exp[N[(c$95$p * N[(N[Log[1 + t$95$1], $MachinePrecision] - N[Log[1 + t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c_p < 2.0000000000000001e-27

   1. Initial program 94.9%

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

   3. Taylor expanded in c_p around 0 99.5%

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

   if 2.0000000000000001e-27 < c_p

   1. Initial program 65.9%

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

   3. Taylor expanded in c_n around 0 68.5%

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

      1. clear-num 68.5%

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

      2. inv-pow 68.5%

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

   5. Applied egg-rr 92.9%

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

      1. unpow-1 92.9%

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

      2. rec-exp 92.9%

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

      3. log1p-undefine 92.9%

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

      4. log-rec 92.9%

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

      5. distribute-lft-out-- 92.9%

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

      6. *-rgt-identity 92.9%

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

      7. cancel-sign-sub 92.9%

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

      8. log-rec 92.9%

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

      9. log1p-undefine 92.9%

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

      10. *-rgt-identity 92.9%

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

   7. Simplified 92.9%

      \[\leadsto \color{blue}{e^{-c\_p \cdot \left(\left(-\mathsf{log1p}\left(e^{-t}\right)\right) + \mathsf{log1p}\left(e^{-s}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{{\left(1 + \frac{1}{-1 - e^{-s}}\right)}^{c\_n}}{{\left(1 + \frac{1}{-1 - e^{-t}}\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;e^{c\_p \cdot \left(\mathsf{log1p}\left(e^{-t}\right) - \mathsf{log1p}\left(e^{-s}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{-s}\\ \mathbf{if}\;-s \leq 100000000:\\ \;\;\;\;\frac{{\left(1 + \frac{1}{-1 - t\_1}\right)}^{c\_n}}{{\left(1 + \frac{1}{-1 - e^{-t}}\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{1 + t\_1}\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (exp (- s))))
   (if (<= (- s) 100000000.0)
     (/
      (pow (+ 1.0 (/ 1.0 (- -1.0 t_1))) c_n)
      (pow (+ 1.0 (/ 1.0 (- -1.0 (exp (- t))))) c_n))
     (pow (/ 1.0 (+ 1.0 t_1)) c_p))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = exp(-s);
	double tmp;
	if (-s <= 100000000.0) {
		tmp = pow((1.0 + (1.0 / (-1.0 - t_1))), c_n) / pow((1.0 + (1.0 / (-1.0 - exp(-t)))), c_n);
	} else {
		tmp = pow((1.0 / (1.0 + t_1)), c_p);
	}
	return tmp;
}

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) :: tmp
    t_1 = exp(-s)
    if (-s <= 100000000.0d0) then
        tmp = ((1.0d0 + (1.0d0 / ((-1.0d0) - t_1))) ** c_n) / ((1.0d0 + (1.0d0 / ((-1.0d0) - exp(-t)))) ** c_n)
    else
        tmp = (1.0d0 / (1.0d0 + t_1)) ** c_p
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = Math.exp(-s);
	double tmp;
	if (-s <= 100000000.0) {
		tmp = Math.pow((1.0 + (1.0 / (-1.0 - t_1))), c_n) / Math.pow((1.0 + (1.0 / (-1.0 - Math.exp(-t)))), c_n);
	} else {
		tmp = Math.pow((1.0 / (1.0 + t_1)), c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	t_1 = math.exp(-s)
	tmp = 0
	if -s <= 100000000.0:
		tmp = math.pow((1.0 + (1.0 / (-1.0 - t_1))), c_n) / math.pow((1.0 + (1.0 / (-1.0 - math.exp(-t)))), c_n)
	else:
		tmp = math.pow((1.0 / (1.0 + t_1)), c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	t_1 = exp(Float64(-s))
	tmp = 0.0
	if (Float64(-s) <= 100000000.0)
		tmp = Float64((Float64(1.0 + Float64(1.0 / Float64(-1.0 - t_1))) ^ c_n) / (Float64(1.0 + Float64(1.0 / Float64(-1.0 - exp(Float64(-t))))) ^ c_n));
	else
		tmp = Float64(1.0 / Float64(1.0 + t_1)) ^ c_p;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	t_1 = exp(-s);
	tmp = 0.0;
	if (-s <= 100000000.0)
		tmp = ((1.0 + (1.0 / (-1.0 - t_1))) ^ c_n) / ((1.0 + (1.0 / (-1.0 - exp(-t)))) ^ c_n);
	else
		tmp = (1.0 / (1.0 + t_1)) ^ c_p;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Exp[(-s)], $MachinePrecision]}, If[LessEqual[(-s), 100000000.0], N[(N[Power[N[(1.0 + N[(1.0 / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision] / N[Power[N[(1.0 + N[(1.0 / N[(-1.0 - N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision], N[Power[N[(1.0 / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 1e8

   1. Initial program 92.0%

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

   3. Taylor expanded in c_p around 0 97.4%

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

   if 1e8 < (neg.f64 s)

   1. Initial program 37.5%

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

   3. Taylor expanded in c_n around 0 37.5%

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

   4. Taylor expanded in c_p around 0 100.0%

      \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq 100000000:\\ \;\;\;\;\frac{{\left(1 + \frac{1}{-1 - e^{-s}}\right)}^{c\_n}}{{\left(1 + \frac{1}{-1 - e^{-t}}\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 50000:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(1 + \frac{1}{-1 - e^{-t}}\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 50000.0)
   (/ (pow 0.5 c_n) (pow (+ 1.0 (/ 1.0 (- -1.0 (exp (- t))))) c_n))
   (pow (/ 1.0 (+ 1.0 (exp (- s)))) c_p)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 50000.0) {
		tmp = pow(0.5, c_n) / pow((1.0 + (1.0 / (-1.0 - exp(-t)))), c_n);
	} else {
		tmp = pow((1.0 / (1.0 + exp(-s))), c_p);
	}
	return tmp;
}

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) :: tmp
    if (-s <= 50000.0d0) then
        tmp = (0.5d0 ** c_n) / ((1.0d0 + (1.0d0 / ((-1.0d0) - exp(-t)))) ** c_n)
    else
        tmp = (1.0d0 / (1.0d0 + exp(-s))) ** c_p
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 50000.0) {
		tmp = Math.pow(0.5, c_n) / Math.pow((1.0 + (1.0 / (-1.0 - Math.exp(-t)))), c_n);
	} else {
		tmp = Math.pow((1.0 / (1.0 + Math.exp(-s))), c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if -s <= 50000.0:
		tmp = math.pow(0.5, c_n) / math.pow((1.0 + (1.0 / (-1.0 - math.exp(-t)))), c_n)
	else:
		tmp = math.pow((1.0 / (1.0 + math.exp(-s))), c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= 50000.0)
		tmp = Float64((0.5 ^ c_n) / (Float64(1.0 + Float64(1.0 / Float64(-1.0 - exp(Float64(-t))))) ^ c_n));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-s)))) ^ c_p;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (-s <= 50000.0)
		tmp = (0.5 ^ c_n) / ((1.0 + (1.0 / (-1.0 - exp(-t)))) ^ c_n);
	else
		tmp = (1.0 / (1.0 + exp(-s))) ^ c_p;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 50000.0], N[(N[Power[0.5, c$95$n], $MachinePrecision] / N[Power[N[(1.0 + N[(1.0 / N[(-1.0 - N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision], N[Power[N[(1.0 / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 5e4

   1. Initial program 92.3%

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

   3. Taylor expanded in c_p around 0 97.4%

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

   4. Taylor expanded in s around 0 96.0%

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

   if 5e4 < (neg.f64 s)

   1. Initial program 33.3%

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

   3. Taylor expanded in c_n around 0 33.3%

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

   4. Taylor expanded in c_p around 0 89.2%

      \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq 50000:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(1 + \frac{1}{-1 - e^{-t}}\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 50000:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(0.5 + t \cdot -0.25\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 50000.0)
   (/ (pow 0.5 c_n) (pow (+ 0.5 (* t -0.25)) c_n))
   (pow (/ 1.0 (+ 1.0 (exp (- s)))) c_p)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 50000.0) {
		tmp = pow(0.5, c_n) / pow((0.5 + (t * -0.25)), c_n);
	} else {
		tmp = pow((1.0 / (1.0 + exp(-s))), c_p);
	}
	return tmp;
}

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) :: tmp
    if (-s <= 50000.0d0) then
        tmp = (0.5d0 ** c_n) / ((0.5d0 + (t * (-0.25d0))) ** c_n)
    else
        tmp = (1.0d0 / (1.0d0 + exp(-s))) ** c_p
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 50000.0) {
		tmp = Math.pow(0.5, c_n) / Math.pow((0.5 + (t * -0.25)), c_n);
	} else {
		tmp = Math.pow((1.0 / (1.0 + Math.exp(-s))), c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if -s <= 50000.0:
		tmp = math.pow(0.5, c_n) / math.pow((0.5 + (t * -0.25)), c_n)
	else:
		tmp = math.pow((1.0 / (1.0 + math.exp(-s))), c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= 50000.0)
		tmp = Float64((0.5 ^ c_n) / (Float64(0.5 + Float64(t * -0.25)) ^ c_n));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-s)))) ^ c_p;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (-s <= 50000.0)
		tmp = (0.5 ^ c_n) / ((0.5 + (t * -0.25)) ^ c_n);
	else
		tmp = (1.0 / (1.0 + exp(-s))) ^ c_p;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 50000.0], N[(N[Power[0.5, c$95$n], $MachinePrecision] / N[Power[N[(0.5 + N[(t * -0.25), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision], N[Power[N[(1.0 / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 5e4

   1. Initial program 92.3%

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

   3. Taylor expanded in c_p around 0 97.4%

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

   4. Taylor expanded in s around 0 96.0%

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

   5. Taylor expanded in t around 0 96.0%

      \[\leadsto \frac{{0.5}^{c\_n}}{{\color{blue}{\left(0.5 + -0.25 \cdot t\right)}}^{c\_n}} \]
   6. Step-by-step derivation

      1. *-commutative 96.0%

         \[\leadsto \frac{{0.5}^{c\_n}}{{\left(0.5 + \color{blue}{t \cdot -0.25}\right)}^{c\_n}} \]

   7. Simplified 96.0%

      \[\leadsto \frac{{0.5}^{c\_n}}{{\color{blue}{\left(0.5 + t \cdot -0.25\right)}}^{c\_n}} \]

   if 5e4 < (neg.f64 s)

   1. Initial program 33.3%

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

   3. Taylor expanded in c_n around 0 33.3%

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

   4. Taylor expanded in c_p around 0 89.2%

      \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq 50000:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(0.5 + t \cdot -0.25\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{{0.5}^{c\_n}}{{\left(0.5 + t \cdot -0.25\right)}^{c\_n}} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (/ (pow 0.5 c_n) (pow (+ 0.5 (* t -0.25)) c_n)))

C

double code(double c_p, double c_n, double t, double s) {
	return pow(0.5, c_n) / pow((0.5 + (t * -0.25)), 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 = (0.5d0 ** c_n) / ((0.5d0 + (t * (-0.25d0))) ** c_n)
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.pow(0.5, c_n) / Math.pow((0.5 + (t * -0.25)), c_n);
}

Python

def code(c_p, c_n, t, s):
	return math.pow(0.5, c_n) / math.pow((0.5 + (t * -0.25)), c_n)

Julia

function code(c_p, c_n, t, s)
	return Float64((0.5 ^ c_n) / (Float64(0.5 + Float64(t * -0.25)) ^ c_n))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = (0.5 ^ c_n) / ((0.5 + (t * -0.25)) ^ c_n);
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(N[Power[0.5, c$95$n], $MachinePrecision] / N[Power[N[(0.5 + N[(t * -0.25), $MachinePrecision]), $MachinePrecision], c$95$n], $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. Add Preprocessing

3. Taylor expanded in c_p around 0 94.4%

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

4. Taylor expanded in s around 0 92.8%

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

5. Taylor expanded in t around 0 92.7%

   \[\leadsto \frac{{0.5}^{c\_n}}{{\color{blue}{\left(0.5 + -0.25 \cdot t\right)}}^{c\_n}} \]
6. Step-by-step derivation

   1. *-commutative 92.7%

      \[\leadsto \frac{{0.5}^{c\_n}}{{\left(0.5 + \color{blue}{t \cdot -0.25}\right)}^{c\_n}} \]

7. Simplified 92.7%

   \[\leadsto \frac{{0.5}^{c\_n}}{{\color{blue}{\left(0.5 + t \cdot -0.25\right)}}^{c\_n}} \]

8. Final simplification 92.7%

   \[\leadsto \frac{{0.5}^{c\_n}}{{\left(0.5 + t \cdot -0.25\right)}^{c\_n}} \]
9. Add Preprocessing

Alternative 6: 93.7% 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. Add Preprocessing

3. Taylor expanded in c_n around 0 89.9%

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

4. Taylor expanded in c_p around 0 90.9%

   \[\leadsto \color{blue}{1} \]

5. Final simplification 90.9%

   \[\leadsto 1 \]
6. Add Preprocessing

Developer target: 96.3% 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 2024053 
(FPCore (c_p c_n t s)
  :name "Harley's example"
  :precision binary64
  :pre (and (< 0.0 c_p) (< 0.0 c_n))

  :herbie-target
  (* (pow (/ (+ 1.0 (exp (- t))) (+ 1.0 (exp (- s)))) c_p) (pow (/ (+ 1.0 (exp t)) (+ 1.0 (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))))