Octave 3.8, jcobi/2

Percentage Accurate: 24.7% → 80.1%

Time: 3.7s

Alternatives: 5

Speedup: 4.3×

• Could not converge on a ground truth

  1. alpha = 1.7533746046869366e+195
  2. beta = -1.7030234586554808e+258
  3. i = -1.7593880419529947e+99

Specification

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 0\]

Math

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \end{array} \]

FPCore

(FPCore (alpha beta i)
  :precision binary64
  (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
  (/
   (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
   2.0)))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Fortran

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * i)
    code = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Initial Program: 24.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \end{array} \]

FPCore

(FPCore (alpha beta i)
  :precision binary64
  (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
  (/
   (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
   2.0)))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Fortran

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * i)
    code = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Alternative 1: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;i \leq 7.8 \cdot 10^{-292}:\\ \;\;\;\;\mathsf{fma}\left(-1, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, \frac{\beta + \alpha}{\mathsf{fma}\left(i, 2, \left(\beta + \alpha\right) - -2\right)}, 1\right) \cdot 0.5\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
  :precision binary64
  (if (<= i 7.8e-292)
  (fma -1.0 0.5 0.5)
  (*
   (fma
    (/ (- beta alpha) (fma i 2.0 (+ beta alpha)))
    (/ (+ beta alpha) (fma i 2.0 (- (+ beta alpha) -2.0)))
    1.0)
   0.5)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 7.8e-292) {
		tmp = fma(-1.0, 0.5, 0.5);
	} else {
		tmp = fma(((beta - alpha) / fma(i, 2.0, (beta + alpha))), ((beta + alpha) / fma(i, 2.0, ((beta + alpha) - -2.0))), 1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 7.8e-292)
		tmp = fma(-1.0, 0.5, 0.5);
	else
		tmp = Float64(fma(Float64(Float64(beta - alpha) / fma(i, 2.0, Float64(beta + alpha))), Float64(Float64(beta + alpha) / fma(i, 2.0, Float64(Float64(beta + alpha) - -2.0))), 1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 7.8e-292], N[(-1.0 * 0.5 + 0.5), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + alpha), $MachinePrecision] / N[(i * 2.0 + N[(N[(beta + alpha), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 7.8e-292

   1. Initial program 24.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   2. Taylor expanded in alpha around inf

      \[\leadsto \frac{\color{blue}{-1} + 1}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 72.6%

      \[\leadsto \frac{\color{blue}{-1} + 1}{2} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 + 1}{2}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{-1 + 1}}{2} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{-1}{2} + \frac{1}{2}} \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{1}{2}} + \frac{1}{2} \]

      5. metadata-eval N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{1}{2}} + \frac{1}{2} \]

      6. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}} \]

      7. lower-fma.f64 72.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1, 0.5, 0.5\right)} \]

   6. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, 0.5, 0.5\right)} \]

   if 7.8e-292 < i

   1. Initial program 24.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right) \cdot \frac{1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right) \cdot \frac{1}{2}} \]

   3. Applied rewrites 21.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\beta - \alpha, \frac{\beta + \alpha}{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) - -2\right) \cdot \mathsf{fma}\left(i, 2, \beta + \alpha\right)}, 1\right) \cdot 0.5} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\beta - \alpha\right) \cdot \frac{\beta + \alpha}{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) - -2\right) \cdot \mathsf{fma}\left(i, 2, \beta + \alpha\right)} + 1\right)} \cdot \frac{1}{2} \]

   5. Applied rewrites 45.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, \frac{\beta + \alpha}{\mathsf{fma}\left(i, 2, \left(\beta + \alpha\right) - -2\right)}, 1\right)} \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 79.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;i \leq 7.8 \cdot 10^{-292}:\\ \;\;\;\;\mathsf{fma}\left(-1, 0.5, 0.5\right)\\ \mathbf{elif}\;i \leq 6.4 \cdot 10^{+233}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha - \beta}{-2 - \left(\beta + \alpha\right)}, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
  :precision binary64
  (if (<= i 7.8e-292)
  (fma -1.0 0.5 0.5)
  (if (<= i 6.4e+233)
    (fma (/ (- alpha beta) (- -2.0 (+ beta alpha))) 0.5 0.5)
    0.5)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 7.8e-292) {
		tmp = fma(-1.0, 0.5, 0.5);
	} else if (i <= 6.4e+233) {
		tmp = fma(((alpha - beta) / (-2.0 - (beta + alpha))), 0.5, 0.5);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 7.8e-292)
		tmp = fma(-1.0, 0.5, 0.5);
	elseif (i <= 6.4e+233)
		tmp = fma(Float64(Float64(alpha - beta) / Float64(-2.0 - Float64(beta + alpha))), 0.5, 0.5);
	else
		tmp = 0.5;
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 7.8e-292], N[(-1.0 * 0.5 + 0.5), $MachinePrecision], If[LessEqual[i, 6.4e+233], N[(N[(N[(alpha - beta), $MachinePrecision] / N[(-2.0 - N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision], 0.5]]

Derivation

1. Split input into 3 regimes

2. if i < 7.8e-292

   1. Initial program 24.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   2. Taylor expanded in alpha around inf

      \[\leadsto \frac{\color{blue}{-1} + 1}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 72.6%

      \[\leadsto \frac{\color{blue}{-1} + 1}{2} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 + 1}{2}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{-1 + 1}}{2} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{-1}{2} + \frac{1}{2}} \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{1}{2}} + \frac{1}{2} \]

      5. metadata-eval N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{1}{2}} + \frac{1}{2} \]

      6. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}} \]

      7. lower-fma.f64 72.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1, 0.5, 0.5\right)} \]

   6. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, 0.5, 0.5\right)} \]

   if 7.8e-292 < i < 6.4000000000000004e233

   1. Initial program 24.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   2. Taylor expanded in i around 0

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{2} + \left(\alpha + \beta\right)} + 1}{2} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\beta - \alpha}{2 + \color{blue}{\left(\alpha + \beta\right)}} + 1}{2} \]

      4. lower-+.f64 47.5%

         \[\leadsto \frac{\frac{\beta - \alpha}{2 + \left(\alpha + \color{blue}{\beta}\right)} + 1}{2} \]

   4. Applied rewrites 47.5%

      \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)} + 1}{2}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)} + 1}}{2} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}}{2} + \frac{1}{2}} \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)} \cdot \frac{1}{2}} + \frac{1}{2} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)} \cdot \color{blue}{\frac{1}{2}} + \frac{1}{2} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}} \]

      7. lower-fma.f64 47.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}, 0.5, 0.5\right)} \]

   6. Applied rewrites 47.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\alpha - \beta}{-2 - \left(\beta + \alpha\right)}, 0.5, 0.5\right)} \]

   if 6.4000000000000004e233 < i

   1. Initial program 24.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   3. Step-by-step derivation

   4. Applied rewrites 19.7%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 73.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;i \leq 1.35 \cdot 10^{+231}:\\ \;\;\;\;\mathsf{fma}\left(-1, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
  :precision binary64
  (if (<= i 1.35e+231) (fma -1.0 0.5 0.5) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1.35e+231) {
		tmp = fma(-1.0, 0.5, 0.5);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 1.35e+231)
		tmp = fma(-1.0, 0.5, 0.5);
	else
		tmp = 0.5;
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 1.35e+231], N[(-1.0 * 0.5 + 0.5), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if i < 1.35e231

   1. Initial program 24.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   2. Taylor expanded in alpha around inf

      \[\leadsto \frac{\color{blue}{-1} + 1}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 72.6%

      \[\leadsto \frac{\color{blue}{-1} + 1}{2} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 + 1}{2}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{-1 + 1}}{2} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{-1}{2} + \frac{1}{2}} \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{1}{2}} + \frac{1}{2} \]

      5. metadata-eval N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{1}{2}} + \frac{1}{2} \]

      6. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}} \]

      7. lower-fma.f64 72.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1, 0.5, 0.5\right)} \]

   6. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, 0.5, 0.5\right)} \]

   if 1.35e231 < i

   1. Initial program 24.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   3. Step-by-step derivation

   4. Applied rewrites 19.7%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 23.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \leq 0.6:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot 0.5\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
  :precision binary64
  (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
  (if (<=
       (/
        (+
         (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0))
         1.0)
        2.0)
       0.6)
    0.5
    (* 2.0 0.5))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0) <= 0.6) {
		tmp = 0.5;
	} else {
		tmp = 2.0 * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0) <= 0.6d0) then
        tmp = 0.5d0
    else
        tmp = 2.0d0 * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0) <= 0.6) {
		tmp = 0.5;
	} else {
		tmp = 2.0 * 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0) <= 0.6:
		tmp = 0.5
	else:
		tmp = 2.0 * 0.5
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0) <= 0.6)
		tmp = 0.5;
	else
		tmp = Float64(2.0 * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0) <= 0.6)
		tmp = 0.5;
	else
		tmp = 2.0 * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 0.6], 0.5, N[(2.0 * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.59999999999999998

   1. Initial program 24.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   3. Step-by-step derivation

   4. Applied rewrites 19.7%

      \[\leadsto \color{blue}{0.5} \]

   if 0.59999999999999998 < (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 24.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right) \cdot \frac{1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right) \cdot \frac{1}{2}} \]

   3. Applied rewrites 21.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\beta - \alpha, \frac{\beta + \alpha}{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) - -2\right) \cdot \mathsf{fma}\left(i, 2, \beta + \alpha\right)}, 1\right) \cdot 0.5} \]

   4. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{2} \cdot 0.5 \]
   5. Step-by-step derivation

   6. Applied rewrites 11.3%

      \[\leadsto \color{blue}{2} \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 19.7% accurate, 42.2× speedup

Math

\[0.5 \]

FPCore

(FPCore (alpha beta i)
  :precision binary64
  0.5)

C

double code(double alpha, double beta, double i) {
	return 0.5;
}

Fortran

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.5d0
end function

Java

public static double code(double alpha, double beta, double i) {
	return 0.5;
}

Python

def code(alpha, beta, i):
	return 0.5

Julia

function code(alpha, beta, i)
	return 0.5
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.5;
end

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 24.7%

   \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

2. Taylor expanded in i around inf

   \[\leadsto \color{blue}{\frac{1}{2}} \]
3. Step-by-step derivation

4. Applied rewrites 19.7%

   \[\leadsto \color{blue}{0.5} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025313 -o setup:search
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))