Octave 3.8, jcobi/2

Percentage Accurate: 63.1% → 97.6%

Time: 5.6s

Alternatives: 8

Speedup: 2.1×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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: 63.1% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \mathsf{fma}\left(2, \beta, 4 \cdot i\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}, \frac{\beta + \alpha}{\left(\left(\left(\alpha + \beta\right) + i\right) + i\right) + 2}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/
         (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
         2.0)
        2e-12)
     (*
      0.5
      (/
       (- (+ beta (* -1.0 beta)) (* -1.0 (+ 2.0 (fma 2.0 beta (* 4.0 i)))))
       alpha))
     (/
      (fma
       (/ (- beta alpha) (fma i 2.0 (+ beta alpha)))
       (/ (+ beta alpha) (+ (+ (+ (+ alpha beta) i) i) 2.0))
       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) / (t_0 + 2.0)) + 1.0) / 2.0) <= 2e-12) {
		tmp = 0.5 * (((beta + (-1.0 * beta)) - (-1.0 * (2.0 + fma(2.0, beta, (4.0 * i))))) / alpha);
	} else {
		tmp = fma(((beta - alpha) / fma(i, 2.0, (beta + alpha))), ((beta + alpha) / ((((alpha + beta) + i) + i) + 2.0)), 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(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0) <= 2e-12)
		tmp = Float64(0.5 * Float64(Float64(Float64(beta + Float64(-1.0 * beta)) - Float64(-1.0 * Float64(2.0 + fma(2.0, beta, Float64(4.0 * i))))) / alpha));
	else
		tmp = Float64(fma(Float64(Float64(beta - alpha) / fma(i, 2.0, Float64(beta + alpha))), Float64(Float64(beta + alpha) / Float64(Float64(Float64(Float64(alpha + beta) + i) + i) + 2.0)), 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[(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], 2e-12], N[(0.5 * N[(N[(N[(beta + N[(-1.0 * beta), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(2.0 + N[(2.0 * beta + N[(4.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + alpha), $MachinePrecision] / N[(N[(N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision] + i), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $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)) < 1.99999999999999996e-12

   1. Initial program 2.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 15.3%

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

      1. unpow1 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 14.1%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. exp-to-pow N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lower-fma.f64 15.0

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

   7. Applied rewrites 15.0%

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

   8. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 90.5

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

   10. Applied rewrites 90.5%

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

   if 1.99999999999999996e-12 < (/.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 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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. count-2-rev N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 99.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

Alternative 2: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{t\_1 + 2} + 1}{2} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \mathsf{fma}\left(2, \beta, 4 \cdot i\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\beta - \alpha, \frac{\frac{\alpha + \beta}{t\_0}}{\mathsf{fma}\left(t\_0, 2, 4\right)}, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ alpha beta))) (t_1 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/
         (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ t_1 2.0)) 1.0)
         2.0)
        2e-12)
     (*
      0.5
      (/
       (- (+ beta (* -1.0 beta)) (* -1.0 (+ 2.0 (fma 2.0 beta (* 4.0 i)))))
       alpha))
     (fma (- beta alpha) (/ (/ (+ alpha beta) t_0) (fma t_0 2.0 4.0)) 0.5))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (alpha + beta));
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((((alpha + beta) * (beta - alpha)) / t_1) / (t_1 + 2.0)) + 1.0) / 2.0) <= 2e-12) {
		tmp = 0.5 * (((beta + (-1.0 * beta)) - (-1.0 * (2.0 + fma(2.0, beta, (4.0 * i))))) / alpha);
	} else {
		tmp = fma((beta - alpha), (((alpha + beta) / t_0) / fma(t_0, 2.0, 4.0)), 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(t_1 + 2.0)) + 1.0) / 2.0) <= 2e-12)
		tmp = Float64(0.5 * Float64(Float64(Float64(beta + Float64(-1.0 * beta)) - Float64(-1.0 * Float64(2.0 + fma(2.0, beta, Float64(4.0 * i))))) / alpha));
	else
		tmp = fma(Float64(beta - alpha), Float64(Float64(Float64(alpha + beta) / t_0) / fma(t_0, 2.0, 4.0)), 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$1), $MachinePrecision] / N[(t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 2e-12], N[(0.5 * N[(N[(N[(beta + N[(-1.0 * beta), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(2.0 + N[(2.0 * beta + N[(4.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(beta - alpha), $MachinePrecision] * N[(N[(N[(alpha + beta), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 * 2.0 + 4.0), $MachinePrecision]), $MachinePrecision] + 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)) < 1.99999999999999996e-12

   1. Initial program 2.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 15.3%

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

      1. unpow1 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 14.1%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. exp-to-pow N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lower-fma.f64 15.0

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

   7. Applied rewrites 15.0%

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

   8. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 90.5

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

   10. Applied rewrites 90.5%

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

   if 1.99999999999999996e-12 < (/.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 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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 99.7%

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

      1. unpow1 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 87.7%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. exp-to-pow N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lower-fma.f64 89.9

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

   7. Applied rewrites 89.9%

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

   9. Applied rewrites 99.6%

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

Alternative 3: 96.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \leq 10^{-6}:\\ \;\;\;\;0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \mathsf{fma}\left(2, \beta, 4 \cdot i\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta - \alpha}{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 2\right) \cdot 2} + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/
         (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
         2.0)
        1e-6)
     (*
      0.5
      (/
       (- (+ beta (* -1.0 beta)) (* -1.0 (+ 2.0 (fma 2.0 beta (* 4.0 i)))))
       alpha))
     (+ (/ (- beta alpha) (* (+ (fma i 2.0 (+ beta alpha)) 2.0) 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) <= 1e-6) {
		tmp = 0.5 * (((beta + (-1.0 * beta)) - (-1.0 * (2.0 + fma(2.0, beta, (4.0 * i))))) / alpha);
	} else {
		tmp = ((beta - alpha) / ((fma(i, 2.0, (beta + alpha)) + 2.0) * 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) <= 1e-6)
		tmp = Float64(0.5 * Float64(Float64(Float64(beta + Float64(-1.0 * beta)) - Float64(-1.0 * Float64(2.0 + fma(2.0, beta, Float64(4.0 * i))))) / alpha));
	else
		tmp = Float64(Float64(Float64(beta - alpha) / Float64(Float64(fma(i, 2.0, Float64(beta + alpha)) + 2.0) * 2.0)) + 0.5);
	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[(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], 1e-6], N[(0.5 * N[(N[(N[(beta + N[(-1.0 * beta), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(2.0 + N[(2.0 * beta + N[(4.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + 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)) < 9.99999999999999955e-7

   1. Initial program 3.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 16.1%

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

      1. unpow1 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 14.7%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. exp-to-pow N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lower-fma.f64 15.7

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

   7. Applied rewrites 15.7%

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

   8. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 90.1

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

   10. Applied rewrites 90.1%

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

   if 9.99999999999999955e-7 < (/.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 81.0%

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

      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. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lower-+.f64 N/A

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

   3. Applied rewrites 81.0%

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

   4. Taylor expanded in i around 0

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

      1. lower--.f64 98.7

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

   6. Applied rewrites 98.7%

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

Alternative 4: 76.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \leq 0.75:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/
         (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
         2.0)
        0.75)
     0.5
     1.0)))

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) / (t_0 + 2.0)) + 1.0) / 2.0) <= 0.75) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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) <= 0.75d0) 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 t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0) <= 0.75) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	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) <= 0.75:
		tmp = 0.5
	else:
		tmp = 1.0
	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(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0) <= 0.75)
		tmp = 0.5;
	else
		tmp = 1.0;
	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) <= 0.75)
		tmp = 0.5;
	else
		tmp = 1.0;
	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], 0.75], 0.5, 1.0]]

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)) < 0.75

   1. Initial program 71.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

   3. Applied rewrites 71.1%

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

      1. unpow1 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 61.3%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 72.4%

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

   if 0.75 < (/.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 34.6%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

   3. Applied rewrites 32.4%

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

      1. unpow1 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 30.4%

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

   6. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 90.7%

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

Alternative 5: 79.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 4.1 \cdot 10^{-50}:\\ \;\;\;\;0.5 \cdot \frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)} + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 2\right) \cdot 2} + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 4.1e-50)
   (+ (* 0.5 (/ (- beta alpha) (+ 2.0 (+ alpha beta)))) 0.5)
   (+ (/ beta (* (+ (fma i 2.0 (+ beta alpha)) 2.0) 2.0)) 0.5)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 4.1e-50) {
		tmp = (0.5 * ((beta - alpha) / (2.0 + (alpha + beta)))) + 0.5;
	} else {
		tmp = (beta / ((fma(i, 2.0, (beta + alpha)) + 2.0) * 2.0)) + 0.5;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 4.1e-50)
		tmp = Float64(Float64(0.5 * Float64(Float64(beta - alpha) / Float64(2.0 + Float64(alpha + beta)))) + 0.5);
	else
		tmp = Float64(Float64(beta / Float64(Float64(fma(i, 2.0, Float64(beta + alpha)) + 2.0) * 2.0)) + 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 4.1e-50], N[(N[(0.5 * N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(beta / N[(N[(N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 4.09999999999999985e-50

   1. Initial program 58.2%

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

      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. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lower-+.f64 N/A

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

   3. Applied rewrites 58.2%

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

   4. Taylor expanded in i around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 73.5

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

   6. Applied rewrites 73.5%

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

   if 4.09999999999999985e-50 < i

   1. Initial program 66.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lower-+.f64 N/A

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

   3. Applied rewrites 66.7%

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

   4. Taylor expanded in beta around inf

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

   6. Applied rewrites 83.5%

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

Alternative 6: 75.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 4.3 \cdot 10^{+42}:\\ \;\;\;\;0.5 \cdot \frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)} + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 4.3e+42)
   (+ (* 0.5 (/ (- beta alpha) (+ 2.0 (+ alpha beta)))) 0.5)
   0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 4.3e+42) {
		tmp = (0.5 * ((beta - alpha) / (2.0 + (alpha + beta)))) + 0.5;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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) :: tmp
    if (i <= 4.3d+42) then
        tmp = (0.5d0 * ((beta - alpha) / (2.0d0 + (alpha + beta)))) + 0.5d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 4.3e+42) {
		tmp = (0.5 * ((beta - alpha) / (2.0 + (alpha + beta)))) + 0.5;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if i <= 4.3e+42:
		tmp = (0.5 * ((beta - alpha) / (2.0 + (alpha + beta)))) + 0.5
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 4.3e+42)
		tmp = Float64(Float64(0.5 * Float64(Float64(beta - alpha) / Float64(2.0 + Float64(alpha + beta)))) + 0.5);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 4.3e+42)
		tmp = (0.5 * ((beta - alpha) / (2.0 + (alpha + beta)))) + 0.5;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 4.3e+42], N[(N[(0.5 * N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if i < 4.2999999999999998e42

   1. Initial program 59.4%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lower-+.f64 N/A

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

   3. Applied rewrites 59.4%

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

   4. Taylor expanded in i around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 73.5

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

   6. Applied rewrites 73.5%

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

   if 4.2999999999999998e42 < i

   1. Initial program 68.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

   3. Applied rewrites 67.4%

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

      1. unpow1 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 76.9%

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

Alternative 7: 79.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{\beta - \alpha}{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 2\right) \cdot 2} + 0.5 \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (+ (/ (- beta alpha) (* (+ (fma i 2.0 (+ beta alpha)) 2.0) 2.0)) 0.5))

C

double code(double alpha, double beta, double i) {
	return ((beta - alpha) / ((fma(i, 2.0, (beta + alpha)) + 2.0) * 2.0)) + 0.5;
}

Julia

function code(alpha, beta, i)
	return Float64(Float64(Float64(beta - alpha) / Float64(Float64(fma(i, 2.0, Float64(beta + alpha)) + 2.0) * 2.0)) + 0.5)
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 63.1%

   \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
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. lift-+.f64 N/A

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

   3. div-add N/A

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

   4. lower-+.f64 N/A

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

3. Applied rewrites 63.1%

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

4. Taylor expanded in i around 0

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

   1. lower--.f64 79.3

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

6. Applied rewrites 79.3%

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

Alternative 8: 61.9% accurate, 73.0× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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 63.1%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. flip-+ N/A

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

   5. associate-/r/ N/A

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

   6. lower-fma.f64 N/A

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

3. Applied rewrites 62.5%

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

   1. unpow1 N/A

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

   2. pow-to-exp N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 54.4%

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

6. Taylor expanded in i around inf

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

8. Applied rewrites 61.9%

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

Reproduce

herbie shell --seed 2025108 
(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))