Octave 3.8, jcobi/2

Percentage Accurate: 61.7% → 97.5%

Time: 14.1s

Alternatives: 10

Speedup: 4.8×

Specification

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

Math

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

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)
    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: 61.7% accurate, 1.0× speedup

Math

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

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)
    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: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.8:\\ \;\;\;\;-0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.8)
     (* -0.5 (+ (* (/ i alpha) -4.0) (/ (- (- -2.0 beta) beta) alpha)))
     (/
      (fma
       (+ alpha beta)
       (/
        (/ (- beta alpha) (+ alpha (+ beta (fma 2.0 i 2.0))))
        (+ alpha (fma 2.0 i beta)))
       1.0)
      2.0))))

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) / (2.0 + t_0)) <= -0.8) {
		tmp = -0.5 * (((i / alpha) * -4.0) + (((-2.0 - beta) - beta) / alpha));
	} else {
		tmp = fma((alpha + beta), (((beta - alpha) / (alpha + (beta + fma(2.0, i, 2.0)))) / (alpha + fma(2.0, i, beta))), 1.0) / 2.0;
	}
	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(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.8)
		tmp = Float64(-0.5 * Float64(Float64(Float64(i / alpha) * -4.0) + Float64(Float64(Float64(-2.0 - beta) - beta) / alpha)));
	else
		tmp = Float64(fma(Float64(alpha + beta), Float64(Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))) / Float64(alpha + fma(2.0, i, beta))), 1.0) / 2.0);
	end
	return 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[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.8], N[(-0.5 * N[(N[(N[(i / alpha), $MachinePrecision] * -4.0), $MachinePrecision] + N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + beta), $MachinePrecision] * N[(N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.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))) < -0.80000000000000004

   1. Initial program 3.0%

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

   3. Simplified 11.2%

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

   5. Taylor expanded in alpha around -inf 94.3%

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

   6. Taylor expanded in i around 0 94.3%

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

      1. associate--l+ 94.3%

         \[\leadsto -0.5 \cdot \color{blue}{\left(-4 \cdot \frac{i}{\alpha} + \left(\left(-1 \cdot \frac{\beta}{\alpha} + \frac{\beta}{\alpha}\right) - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)} \]

      2. *-commutative 94.3%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\frac{i}{\alpha} \cdot -4} + \left(\left(-1 \cdot \frac{\beta}{\alpha} + \frac{\beta}{\alpha}\right) - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      3. distribute-lft1-in 94.3%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{\left(-1 + 1\right) \cdot \frac{\beta}{\alpha}} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      4. metadata-eval 94.3%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      5. mul0-lft 94.3%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{\left(0 \cdot \beta\right)} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      6. mul0-lft 94.3%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      7. mul0-lft 94.3%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{0} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      8. neg-sub0 94.3%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \color{blue}{\left(--1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)}\right) \]

      9. mul-1-neg 94.3%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(-\color{blue}{\left(-\frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)}\right)\right) \]

      10. remove-double-neg 94.3%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \color{blue}{\frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}}\right) \]

      11. neg-mul-1 94.3%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{\left(-\beta\right)} + -1 \cdot \left(2 + \beta\right)}{\alpha}\right) \]

      12. +-commutative 94.3%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{-1 \cdot \left(2 + \beta\right) + \left(-\beta\right)}}{\alpha}\right) \]

      13. unsub-neg 94.3%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{-1 \cdot \left(2 + \beta\right) - \beta}}{\alpha}\right) \]

      14. distribute-lft-in 94.3%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{\left(-1 \cdot 2 + -1 \cdot \beta\right)} - \beta}{\alpha}\right) \]

      15. metadata-eval 94.3%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(\color{blue}{-2} + -1 \cdot \beta\right) - \beta}{\alpha}\right) \]

      16. neg-mul-1 94.3%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(-2 + \color{blue}{\left(-\beta\right)}\right) - \beta}{\alpha}\right) \]

      17. unsub-neg 94.3%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{\left(-2 - \beta\right)} - \beta}{\alpha}\right) \]

   8. Simplified 94.3%

      \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{i}{\alpha} \cdot -4 + \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)} \]

   if -0.80000000000000004 < (/.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)))

   1. Initial program 80.9%

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

   3. Simplified 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.8:\\ \;\;\;\;-0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.8:\\ \;\;\;\;-0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.8)
     (* -0.5 (+ (* (/ i alpha) -4.0) (/ (- (- -2.0 beta) beta) alpha)))
     (/
      (+
       1.0
       (/
        (* (- beta alpha) (/ (+ alpha beta) (fma 2.0 i (+ alpha beta))))
        (+ alpha (+ beta (fma 2.0 i 2.0)))))
      2.0))))

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) / (2.0 + t_0)) <= -0.8) {
		tmp = -0.5 * (((i / alpha) * -4.0) + (((-2.0 - beta) - beta) / alpha));
	} else {
		tmp = (1.0 + (((beta - alpha) * ((alpha + beta) / fma(2.0, i, (alpha + beta)))) / (alpha + (beta + fma(2.0, i, 2.0))))) / 2.0;
	}
	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(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.8)
		tmp = Float64(-0.5 * Float64(Float64(Float64(i / alpha) * -4.0) + Float64(Float64(Float64(-2.0 - beta) - beta) / alpha)));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) / fma(2.0, i, Float64(alpha + beta)))) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0))))) / 2.0);
	end
	return 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[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.8], N[(-0.5 * N[(N[(N[(i / alpha), $MachinePrecision] * -4.0), $MachinePrecision] + N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.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))) < -0.80000000000000004

   1. Initial program 3.0%

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

   3. Simplified 11.2%

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

   5. Taylor expanded in alpha around -inf 94.3%

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

   6. Taylor expanded in i around 0 94.3%

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

      1. associate--l+ 94.3%

         \[\leadsto -0.5 \cdot \color{blue}{\left(-4 \cdot \frac{i}{\alpha} + \left(\left(-1 \cdot \frac{\beta}{\alpha} + \frac{\beta}{\alpha}\right) - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)} \]

      2. *-commutative 94.3%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\frac{i}{\alpha} \cdot -4} + \left(\left(-1 \cdot \frac{\beta}{\alpha} + \frac{\beta}{\alpha}\right) - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      3. distribute-lft1-in 94.3%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{\left(-1 + 1\right) \cdot \frac{\beta}{\alpha}} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      4. metadata-eval 94.3%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      5. mul0-lft 94.3%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{\left(0 \cdot \beta\right)} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      6. mul0-lft 94.3%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      7. mul0-lft 94.3%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{0} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      8. neg-sub0 94.3%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \color{blue}{\left(--1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)}\right) \]

      9. mul-1-neg 94.3%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(-\color{blue}{\left(-\frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)}\right)\right) \]

      10. remove-double-neg 94.3%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \color{blue}{\frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}}\right) \]

      11. neg-mul-1 94.3%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{\left(-\beta\right)} + -1 \cdot \left(2 + \beta\right)}{\alpha}\right) \]

      12. +-commutative 94.3%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{-1 \cdot \left(2 + \beta\right) + \left(-\beta\right)}}{\alpha}\right) \]

      13. unsub-neg 94.3%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{-1 \cdot \left(2 + \beta\right) - \beta}}{\alpha}\right) \]

      14. distribute-lft-in 94.3%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{\left(-1 \cdot 2 + -1 \cdot \beta\right)} - \beta}{\alpha}\right) \]

      15. metadata-eval 94.3%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(\color{blue}{-2} + -1 \cdot \beta\right) - \beta}{\alpha}\right) \]

      16. neg-mul-1 94.3%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(-2 + \color{blue}{\left(-\beta\right)}\right) - \beta}{\alpha}\right) \]

      17. unsub-neg 94.3%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{\left(-2 - \beta\right)} - \beta}{\alpha}\right) \]

   8. Simplified 94.3%

      \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{i}{\alpha} \cdot -4 + \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)} \]

   if -0.80000000000000004 < (/.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)))

   1. Initial program 80.9%

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

   3. Simplified 99.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.8:\\ \;\;\;\;-0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.5:\\ \;\;\;\;-0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.5)
     (* -0.5 (+ (* (/ i alpha) -4.0) (/ (- (- -2.0 beta) beta) alpha)))
     (/
      (+
       1.0
       (/
        (* (- beta alpha) (/ beta (+ beta (* 2.0 i))))
        (+ alpha (+ beta (fma 2.0 i 2.0)))))
      2.0))))

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) / (2.0 + t_0)) <= -0.5) {
		tmp = -0.5 * (((i / alpha) * -4.0) + (((-2.0 - beta) - beta) / alpha));
	} else {
		tmp = (1.0 + (((beta - alpha) * (beta / (beta + (2.0 * i)))) / (alpha + (beta + fma(2.0, i, 2.0))))) / 2.0;
	}
	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(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.5)
		tmp = Float64(-0.5 * Float64(Float64(Float64(i / alpha) * -4.0) + Float64(Float64(Float64(-2.0 - beta) - beta) / alpha)));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta - alpha) * Float64(beta / Float64(beta + Float64(2.0 * i)))) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0))))) / 2.0);
	end
	return 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[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.5], N[(-0.5 * N[(N[(N[(i / alpha), $MachinePrecision] * -4.0), $MachinePrecision] + N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta / N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.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))) < -0.5

   1. Initial program 4.3%

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

   3. Simplified 12.4%

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

   5. Taylor expanded in alpha around -inf 93.2%

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

   6. Taylor expanded in i around 0 93.2%

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

      1. associate--l+ 93.2%

         \[\leadsto -0.5 \cdot \color{blue}{\left(-4 \cdot \frac{i}{\alpha} + \left(\left(-1 \cdot \frac{\beta}{\alpha} + \frac{\beta}{\alpha}\right) - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)} \]

      2. *-commutative 93.2%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\frac{i}{\alpha} \cdot -4} + \left(\left(-1 \cdot \frac{\beta}{\alpha} + \frac{\beta}{\alpha}\right) - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      3. distribute-lft1-in 93.2%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{\left(-1 + 1\right) \cdot \frac{\beta}{\alpha}} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      4. metadata-eval 93.2%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      5. mul0-lft 93.2%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{\left(0 \cdot \beta\right)} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      6. mul0-lft 93.2%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      7. mul0-lft 93.2%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{0} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      8. neg-sub0 93.2%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \color{blue}{\left(--1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)}\right) \]

      9. mul-1-neg 93.2%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(-\color{blue}{\left(-\frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)}\right)\right) \]

      10. remove-double-neg 93.2%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \color{blue}{\frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}}\right) \]

      11. neg-mul-1 93.2%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{\left(-\beta\right)} + -1 \cdot \left(2 + \beta\right)}{\alpha}\right) \]

      12. +-commutative 93.2%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{-1 \cdot \left(2 + \beta\right) + \left(-\beta\right)}}{\alpha}\right) \]

      13. unsub-neg 93.2%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{-1 \cdot \left(2 + \beta\right) - \beta}}{\alpha}\right) \]

      14. distribute-lft-in 93.2%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{\left(-1 \cdot 2 + -1 \cdot \beta\right)} - \beta}{\alpha}\right) \]

      15. metadata-eval 93.2%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(\color{blue}{-2} + -1 \cdot \beta\right) - \beta}{\alpha}\right) \]

      16. neg-mul-1 93.2%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(-2 + \color{blue}{\left(-\beta\right)}\right) - \beta}{\alpha}\right) \]

      17. unsub-neg 93.2%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{\left(-2 - \beta\right)} - \beta}{\alpha}\right) \]

   8. Simplified 93.2%

      \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{i}{\alpha} \cdot -4 + \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)} \]

   if -0.5 < (/.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)))

   1. Initial program 80.8%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 98.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;-0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{2 + t\_1} \leq -0.5:\\ \;\;\;\;-0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{t\_0}}{2 + t\_0}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i))) (t_1 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ 2.0 t_1)) -0.5)
     (* -0.5 (+ (* (/ i alpha) -4.0) (/ (- (- -2.0 beta) beta) alpha)))
     (/ (+ 1.0 (/ (* (- beta alpha) (/ beta t_0)) (+ 2.0 t_0))) 2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.5) {
		tmp = -0.5 * (((i / alpha) * -4.0) + (((-2.0 - beta) - beta) / alpha));
	} else {
		tmp = (1.0 + (((beta - alpha) * (beta / t_0)) / (2.0 + t_0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = beta + (2.0d0 * i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0d0 + t_1)) <= (-0.5d0)) then
        tmp = (-0.5d0) * (((i / alpha) * (-4.0d0)) + ((((-2.0d0) - beta) - beta) / alpha))
    else
        tmp = (1.0d0 + (((beta - alpha) * (beta / t_0)) / (2.0d0 + t_0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.5) {
		tmp = -0.5 * (((i / alpha) * -4.0) + (((-2.0 - beta) - beta) / alpha));
	} else {
		tmp = (1.0 + (((beta - alpha) * (beta / t_0)) / (2.0 + t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (2.0 * i)
	t_1 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.5:
		tmp = -0.5 * (((i / alpha) * -4.0) + (((-2.0 - beta) - beta) / alpha))
	else:
		tmp = (1.0 + (((beta - alpha) * (beta / t_0)) / (2.0 + t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(2.0 + t_1)) <= -0.5)
		tmp = Float64(-0.5 * Float64(Float64(Float64(i / alpha) * -4.0) + Float64(Float64(Float64(-2.0 - beta) - beta) / alpha)));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta - alpha) * Float64(beta / t_0)) / Float64(2.0 + t_0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (2.0 * i);
	t_1 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.5)
		tmp = -0.5 * (((i / alpha) * -4.0) + (((-2.0 - beta) - beta) / alpha));
	else
		tmp = (1.0 + (((beta - alpha) * (beta / t_0)) / (2.0 + t_0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], -0.5], N[(-0.5 * N[(N[(N[(i / alpha), $MachinePrecision] * -4.0), $MachinePrecision] + N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.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))) < -0.5

   1. Initial program 4.3%

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

   3. Simplified 12.4%

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

   5. Taylor expanded in alpha around -inf 93.2%

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

   6. Taylor expanded in i around 0 93.2%

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

      1. associate--l+ 93.2%

         \[\leadsto -0.5 \cdot \color{blue}{\left(-4 \cdot \frac{i}{\alpha} + \left(\left(-1 \cdot \frac{\beta}{\alpha} + \frac{\beta}{\alpha}\right) - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)} \]

      2. *-commutative 93.2%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\frac{i}{\alpha} \cdot -4} + \left(\left(-1 \cdot \frac{\beta}{\alpha} + \frac{\beta}{\alpha}\right) - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      3. distribute-lft1-in 93.2%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{\left(-1 + 1\right) \cdot \frac{\beta}{\alpha}} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      4. metadata-eval 93.2%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      5. mul0-lft 93.2%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{\left(0 \cdot \beta\right)} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      6. mul0-lft 93.2%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      7. mul0-lft 93.2%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{0} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      8. neg-sub0 93.2%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \color{blue}{\left(--1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)}\right) \]

      9. mul-1-neg 93.2%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(-\color{blue}{\left(-\frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)}\right)\right) \]

      10. remove-double-neg 93.2%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \color{blue}{\frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}}\right) \]

      11. neg-mul-1 93.2%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{\left(-\beta\right)} + -1 \cdot \left(2 + \beta\right)}{\alpha}\right) \]

      12. +-commutative 93.2%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{-1 \cdot \left(2 + \beta\right) + \left(-\beta\right)}}{\alpha}\right) \]

      13. unsub-neg 93.2%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{-1 \cdot \left(2 + \beta\right) - \beta}}{\alpha}\right) \]

      14. distribute-lft-in 93.2%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{\left(-1 \cdot 2 + -1 \cdot \beta\right)} - \beta}{\alpha}\right) \]

      15. metadata-eval 93.2%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(\color{blue}{-2} + -1 \cdot \beta\right) - \beta}{\alpha}\right) \]

      16. neg-mul-1 93.2%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(-2 + \color{blue}{\left(-\beta\right)}\right) - \beta}{\alpha}\right) \]

      17. unsub-neg 93.2%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{\left(-2 - \beta\right)} - \beta}{\alpha}\right) \]

   8. Simplified 93.2%

      \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{i}{\alpha} \cdot -4 + \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)} \]

   if -0.5 < (/.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)))

   1. Initial program 80.8%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 98.8%

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

   6. Taylor expanded in alpha around 0 98.6%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;-0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{2 + \left(\beta + 2 \cdot i\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.07 \cdot 10^{+58}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.07e+58)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (* -0.5 (+ (* (/ i alpha) -4.0) (/ (- (- -2.0 beta) beta) alpha)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.07e+58) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = -0.5 * (((i / alpha) * -4.0) + (((-2.0 - beta) - beta) / alpha));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 2.07d+58) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (-0.5d0) * (((i / alpha) * (-4.0d0)) + ((((-2.0d0) - beta) - beta) / alpha))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.07e+58) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = -0.5 * (((i / alpha) * -4.0) + (((-2.0 - beta) - beta) / alpha));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.07e+58:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = -0.5 * (((i / alpha) * -4.0) + (((-2.0 - beta) - beta) / alpha))
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.07e+58)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(-0.5 * Float64(Float64(Float64(i / alpha) * -4.0) + Float64(Float64(Float64(-2.0 - beta) - beta) / alpha)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.07e+58)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = -0.5 * (((i / alpha) * -4.0) + (((-2.0 - beta) - beta) / alpha));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.07e+58], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(-0.5 * N[(N[(N[(i / alpha), $MachinePrecision] * -4.0), $MachinePrecision] + N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.07000000000000011e58

   1. Initial program 79.6%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in i around 0 86.8%

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

   6. Taylor expanded in alpha around 0 88.0%

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

   if 2.07000000000000011e58 < alpha

   1. Initial program 14.2%

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

   3. Simplified 22.9%

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

   5. Taylor expanded in alpha around -inf 80.1%

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

   6. Taylor expanded in i around 0 80.1%

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

      1. associate--l+ 80.1%

         \[\leadsto -0.5 \cdot \color{blue}{\left(-4 \cdot \frac{i}{\alpha} + \left(\left(-1 \cdot \frac{\beta}{\alpha} + \frac{\beta}{\alpha}\right) - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)} \]

      2. *-commutative 80.1%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\frac{i}{\alpha} \cdot -4} + \left(\left(-1 \cdot \frac{\beta}{\alpha} + \frac{\beta}{\alpha}\right) - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      3. distribute-lft1-in 80.1%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{\left(-1 + 1\right) \cdot \frac{\beta}{\alpha}} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      4. metadata-eval 80.1%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      5. mul0-lft 80.1%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{\left(0 \cdot \beta\right)} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      6. mul0-lft 80.1%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      7. mul0-lft 80.1%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{0} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      8. neg-sub0 80.1%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \color{blue}{\left(--1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)}\right) \]

      9. mul-1-neg 80.1%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(-\color{blue}{\left(-\frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)}\right)\right) \]

      10. remove-double-neg 80.1%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \color{blue}{\frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}}\right) \]

      11. neg-mul-1 80.1%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{\left(-\beta\right)} + -1 \cdot \left(2 + \beta\right)}{\alpha}\right) \]

      12. +-commutative 80.1%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{-1 \cdot \left(2 + \beta\right) + \left(-\beta\right)}}{\alpha}\right) \]

      13. unsub-neg 80.1%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{-1 \cdot \left(2 + \beta\right) - \beta}}{\alpha}\right) \]

      14. distribute-lft-in 80.1%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{\left(-1 \cdot 2 + -1 \cdot \beta\right)} - \beta}{\alpha}\right) \]

      15. metadata-eval 80.1%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(\color{blue}{-2} + -1 \cdot \beta\right) - \beta}{\alpha}\right) \]

      16. neg-mul-1 80.1%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(-2 + \color{blue}{\left(-\beta\right)}\right) - \beta}{\alpha}\right) \]

      17. unsub-neg 80.1%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{\left(-2 - \beta\right)} - \beta}{\alpha}\right) \]

   8. Simplified 80.1%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.07 \cdot 10^{+58}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.07 \cdot 10^{+58}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{i \cdot -4}{\alpha} - \frac{2}{\alpha}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.07e+58)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (* -0.5 (- (/ (* i -4.0) alpha) (/ 2.0 alpha)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.07e+58) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = -0.5 * (((i * -4.0) / alpha) - (2.0 / alpha));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 2.07d+58) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (-0.5d0) * (((i * (-4.0d0)) / alpha) - (2.0d0 / alpha))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.07e+58) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = -0.5 * (((i * -4.0) / alpha) - (2.0 / alpha));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.07e+58:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = -0.5 * (((i * -4.0) / alpha) - (2.0 / alpha))
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.07e+58)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(-0.5 * Float64(Float64(Float64(i * -4.0) / alpha) - Float64(2.0 / alpha)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.07e+58)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = -0.5 * (((i * -4.0) / alpha) - (2.0 / alpha));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.07e+58], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(-0.5 * N[(N[(N[(i * -4.0), $MachinePrecision] / alpha), $MachinePrecision] - N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.07000000000000011e58

   1. Initial program 79.6%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in i around 0 86.8%

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

   6. Taylor expanded in alpha around 0 88.0%

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

   if 2.07000000000000011e58 < alpha

   1. Initial program 14.2%

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

   3. Simplified 22.9%

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

   5. Taylor expanded in alpha around -inf 80.1%

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

   6. Taylor expanded in i around 0 80.1%

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

      1. associate--l+ 80.1%

         \[\leadsto -0.5 \cdot \color{blue}{\left(-4 \cdot \frac{i}{\alpha} + \left(\left(-1 \cdot \frac{\beta}{\alpha} + \frac{\beta}{\alpha}\right) - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right)} \]

      2. *-commutative 80.1%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\frac{i}{\alpha} \cdot -4} + \left(\left(-1 \cdot \frac{\beta}{\alpha} + \frac{\beta}{\alpha}\right) - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      3. distribute-lft1-in 80.1%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{\left(-1 + 1\right) \cdot \frac{\beta}{\alpha}} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      4. metadata-eval 80.1%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      5. mul0-lft 80.1%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{\left(0 \cdot \beta\right)} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      6. mul0-lft 80.1%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{0} \cdot \frac{\beta}{\alpha} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      7. mul0-lft 80.1%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(\color{blue}{0} - -1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)\right) \]

      8. neg-sub0 80.1%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \color{blue}{\left(--1 \cdot \frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)}\right) \]

      9. mul-1-neg 80.1%

         \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \left(-\color{blue}{\left(-\frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}\right)}\right)\right) \]

      10. remove-double-neg 80.1%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \color{blue}{\frac{-1 \cdot \beta + -1 \cdot \left(2 + \beta\right)}{\alpha}}\right) \]

      11. neg-mul-1 80.1%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{\left(-\beta\right)} + -1 \cdot \left(2 + \beta\right)}{\alpha}\right) \]

      12. +-commutative 80.1%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{-1 \cdot \left(2 + \beta\right) + \left(-\beta\right)}}{\alpha}\right) \]

      13. unsub-neg 80.1%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{-1 \cdot \left(2 + \beta\right) - \beta}}{\alpha}\right) \]

      14. distribute-lft-in 80.1%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{\left(-1 \cdot 2 + -1 \cdot \beta\right)} - \beta}{\alpha}\right) \]

      15. metadata-eval 80.1%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(\color{blue}{-2} + -1 \cdot \beta\right) - \beta}{\alpha}\right) \]

      16. neg-mul-1 80.1%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\left(-2 + \color{blue}{\left(-\beta\right)}\right) - \beta}{\alpha}\right) \]

      17. unsub-neg 80.1%

          \[\leadsto -0.5 \cdot \left(\frac{i}{\alpha} \cdot -4 + \frac{\color{blue}{\left(-2 - \beta\right)} - \beta}{\alpha}\right) \]

   8. Simplified 80.1%

      \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{i}{\alpha} \cdot -4 + \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)} \]

   9. Taylor expanded in beta around 0 68.8%

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

       1. *-commutative 68.8%

          \[\leadsto -0.5 \cdot \left(\color{blue}{\frac{i}{\alpha} \cdot -4} - 2 \cdot \frac{1}{\alpha}\right) \]

       2. associate-*l/ 68.8%

          \[\leadsto -0.5 \cdot \left(\color{blue}{\frac{i \cdot -4}{\alpha}} - 2 \cdot \frac{1}{\alpha}\right) \]

       3. associate-*r/ 68.8%

          \[\leadsto -0.5 \cdot \left(\frac{i \cdot -4}{\alpha} - \color{blue}{\frac{2 \cdot 1}{\alpha}}\right) \]

       4. metadata-eval 68.8%

          \[\leadsto -0.5 \cdot \left(\frac{i \cdot -4}{\alpha} - \frac{\color{blue}{2}}{\alpha}\right) \]

   11. Simplified 68.8%

       \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{i \cdot -4}{\alpha} - \frac{2}{\alpha}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.07 \cdot 10^{+58}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{i \cdot -4}{\alpha} - \frac{2}{\alpha}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.07 \cdot 10^{+58}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.07e+58)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.07e+58) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 2.07d+58) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.07e+58) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.07e+58:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.07e+58)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.07e+58)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.07e+58], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.07000000000000011e58

   1. Initial program 79.6%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in i around 0 86.8%

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

   6. Taylor expanded in alpha around 0 88.0%

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

   if 2.07000000000000011e58 < alpha

   1. Initial program 14.2%

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

   3. Simplified 25.2%

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

   5. Taylor expanded in i around 0 8.8%

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

   6. Taylor expanded in alpha around inf 57.9%

      \[\leadsto \frac{\color{blue}{\frac{2 + 2 \cdot \beta}{\alpha}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.07 \cdot 10^{+58}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.07 \cdot 10^{+58}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.07e+58)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (+ (/ beta alpha) (/ 1.0 alpha))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.07e+58) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta / alpha) + (1.0 / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 2.07d+58) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (beta / alpha) + (1.0d0 / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.07e+58) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta / alpha) + (1.0 / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.07e+58:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (beta / alpha) + (1.0 / alpha)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.07e+58)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(beta / alpha) + Float64(1.0 / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.07e+58)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (beta / alpha) + (1.0 / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.07e+58], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta / alpha), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.07000000000000011e58

   1. Initial program 79.6%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in i around 0 86.8%

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

   6. Taylor expanded in alpha around 0 88.0%

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

   if 2.07000000000000011e58 < alpha

   1. Initial program 14.2%

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

   3. Simplified 25.2%

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

   5. Taylor expanded in i around 0 8.8%

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

   6. Taylor expanded in alpha around inf 57.9%

      \[\leadsto \frac{\color{blue}{\frac{2 + 2 \cdot \beta}{\alpha}}}{2} \]

   7. Taylor expanded in beta around 0 57.9%

      \[\leadsto \color{blue}{\frac{1}{\alpha} + \frac{\beta}{\alpha}} \]
   8. Step-by-step derivation

      1. +-commutative 57.9%

         \[\leadsto \color{blue}{\frac{\beta}{\alpha} + \frac{1}{\alpha}} \]

   9. Simplified 57.9%

      \[\leadsto \color{blue}{\frac{\beta}{\alpha} + \frac{1}{\alpha}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.07 \cdot 10^{+58}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9 \cdot 10^{+70}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 9e+70) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9e+70) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 9d+70) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9e+70) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 9e+70:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 9e+70)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 9e+70)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 9e+70], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 8.9999999999999999e70

   1. Initial program 68.2%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in i around inf 66.3%

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

   if 8.9999999999999999e70 < beta

   1. Initial program 38.9%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in i around 0 77.4%

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

   6. Taylor expanded in beta around inf 78.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 60.5% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

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)
    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 60.2%

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

3. Simplified 65.9%

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

5. Taylor expanded in i around inf 56.0%

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

Reproduce

herbie shell --seed 2024141 
(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))