Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B

Percentage Accurate: 75.2% → 87.7%

Time: 15.9s

Alternatives: 12

Speedup: 0.6×

Specification

Math

\[\begin{array}{l} \\ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x + ((y * z) / t)) / ((a + 1.0d0) + ((y * b) / t))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

Python

def code(x, y, z, t, a, b):
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x + ((y * z) / t)) / ((a + 1.0d0) + ((y * b) / t))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

Python

def code(x, y, z, t, a, b):
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 87.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot b}{t}\\ t_2 := \frac{x + \frac{y \cdot z}{t}}{t\_1 + \left(a + 1\right)}\\ t_3 := 1 + \left(a + t\_1\right)\\ t_4 := 1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-286}:\\ \;\;\;\;\frac{x}{t\_3} + \frac{y \cdot z}{t \cdot t\_3}\\ \mathbf{elif}\;t\_2 \leq 0 \lor \neg \left(t\_2 \leq \infty\right):\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, \frac{z}{t\_4}, \frac{x}{t\_4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* y b) t))
        (t_2 (/ (+ x (/ (* y z) t)) (+ t_1 (+ a 1.0))))
        (t_3 (+ 1.0 (+ a t_1)))
        (t_4 (+ 1.0 (fma b (/ y t) a))))
   (if (<= t_2 -5e-286)
     (+ (/ x t_3) (/ (* y z) (* t t_3)))
     (if (or (<= t_2 0.0) (not (<= t_2 INFINITY)))
       (/ (+ z (/ (* x t) y)) b)
       (fma (/ y t) (/ z t_4) (/ x t_4))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * b) / t;
	double t_2 = (x + ((y * z) / t)) / (t_1 + (a + 1.0));
	double t_3 = 1.0 + (a + t_1);
	double t_4 = 1.0 + fma(b, (y / t), a);
	double tmp;
	if (t_2 <= -5e-286) {
		tmp = (x / t_3) + ((y * z) / (t * t_3));
	} else if ((t_2 <= 0.0) || !(t_2 <= ((double) INFINITY))) {
		tmp = (z + ((x * t) / y)) / b;
	} else {
		tmp = fma((y / t), (z / t_4), (x / t_4));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * b) / t)
	t_2 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(t_1 + Float64(a + 1.0)))
	t_3 = Float64(1.0 + Float64(a + t_1))
	t_4 = Float64(1.0 + fma(b, Float64(y / t), a))
	tmp = 0.0
	if (t_2 <= -5e-286)
		tmp = Float64(Float64(x / t_3) + Float64(Float64(y * z) / Float64(t * t_3)));
	elseif ((t_2 <= 0.0) || !(t_2 <= Inf))
		tmp = Float64(Float64(z + Float64(Float64(x * t) / y)) / b);
	else
		tmp = fma(Float64(y / t), Float64(z / t_4), Float64(x / t_4));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(a + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 + N[(b * N[(y / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-286], N[(N[(x / t$95$3), $MachinePrecision] + N[(N[(y * z), $MachinePrecision] / N[(t * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$2, 0.0], N[Not[LessEqual[t$95$2, Infinity]], $MachinePrecision]], N[(N[(z + N[(N[(x * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * N[(z / t$95$4), $MachinePrecision] + N[(x / t$95$4), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -5.00000000000000037e-286

   1. Initial program 85.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. +-commutative 85.0%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 89.0%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. fma-define 89.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative 89.0%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]

      5. associate-/l* 85.6%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{y \cdot \frac{b}{t}} + \left(a + 1\right)} \]

      6. fma-define 85.6%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]

   3. Simplified 85.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 91.7%

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} + \frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}} \]

   if -5.00000000000000037e-286 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0 or +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 30.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. +-commutative 30.6%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 32.2%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. fma-define 32.2%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative 32.2%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]

      5. associate-/l* 40.1%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{y \cdot \frac{b}{t}} + \left(a + 1\right)} \]

      6. fma-define 40.1%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]

   3. Simplified 40.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 32.8%

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} + \frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 32.8%

         \[\leadsto \color{blue}{\frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]

      2. times-frac 25.9%

         \[\leadsto \color{blue}{\frac{y}{t} \cdot \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} \]

      3. fma-define 25.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right)} \]

      4. associate-*r/ 32.1%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      5. +-commutative 32.1%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      6. fma-define 32.1%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      7. associate-*r/ 36.8%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}\right) \]

      8. +-commutative 36.8%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}\right) \]

      9. fma-define 36.8%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}\right) \]

   7. Simplified 36.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\right)} \]

   8. Taylor expanded in b around inf 82.7%

      \[\leadsto \color{blue}{\frac{z + \frac{t \cdot x}{y}}{b}} \]

   if 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 85.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. +-commutative 85.6%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 85.6%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. fma-define 85.6%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative 85.6%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]

      5. associate-/l* 80.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{y \cdot \frac{b}{t}} + \left(a + 1\right)} \]

      6. fma-define 80.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]

   3. Simplified 80.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 88.1%

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} + \frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 88.1%

         \[\leadsto \color{blue}{\frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]

      2. times-frac 93.8%

         \[\leadsto \color{blue}{\frac{y}{t} \cdot \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} \]

      3. fma-define 93.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right)} \]

      4. associate-*r/ 92.4%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      5. +-commutative 92.4%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      6. fma-define 92.4%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      7. associate-*r/ 92.4%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}\right) \]

      8. +-commutative 92.4%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}\right) \]

      9. fma-define 92.4%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}\right) \]

   7. Simplified 92.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -5 \cdot 10^{-286}:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)} + \frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{y \cdot b}{t}\right)\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0 \lor \neg \left(\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq \infty\right):\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-286} \lor \neg \left(t\_1 \leq 0\right) \land t\_1 \leq 4 \cdot 10^{+280}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (/ (* y b) t) (+ a 1.0)))))
   (if (<= t_1 (- INFINITY))
     (/ (* y z) (+ (* y b) (* t (+ a 1.0))))
     (if (or (<= t_1 -5e-286) (and (not (<= t_1 0.0)) (<= t_1 4e+280)))
       t_1
       (/ (+ z (/ (* x t) y)) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (y * z) / ((y * b) + (t * (a + 1.0)));
	} else if ((t_1 <= -5e-286) || (!(t_1 <= 0.0) && (t_1 <= 4e+280))) {
		tmp = t_1;
	} else {
		tmp = (z + ((x * t) / y)) / b;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (y * z) / ((y * b) + (t * (a + 1.0)));
	} else if ((t_1 <= -5e-286) || (!(t_1 <= 0.0) && (t_1 <= 4e+280))) {
		tmp = t_1;
	} else {
		tmp = (z + ((x * t) / y)) / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (y * z) / ((y * b) + (t * (a + 1.0)))
	elif (t_1 <= -5e-286) or (not (t_1 <= 0.0) and (t_1 <= 4e+280)):
		tmp = t_1
	else:
		tmp = (z + ((x * t) / y)) / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(y * z) / Float64(Float64(y * b) + Float64(t * Float64(a + 1.0))));
	elseif ((t_1 <= -5e-286) || (!(t_1 <= 0.0) && (t_1 <= 4e+280)))
		tmp = t_1;
	else
		tmp = Float64(Float64(z + Float64(Float64(x * t) / y)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (y * z) / ((y * b) + (t * (a + 1.0)));
	elseif ((t_1 <= -5e-286) || (~((t_1 <= 0.0)) && (t_1 <= 4e+280)))
		tmp = t_1;
	else
		tmp = (z + ((x * t) / y)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y * z), $MachinePrecision] / N[(N[(y * b), $MachinePrecision] + N[(t * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$1, -5e-286], And[N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision], LessEqual[t$95$1, 4e+280]]], t$95$1, N[(N[(z + N[(N[(x * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0

   1. Initial program 30.2%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 30.2%

         \[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 55.0%

         \[\leadsto \frac{x + \color{blue}{z \cdot \frac{y}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   4. Applied egg-rr 55.0%

      \[\leadsto \frac{x + \color{blue}{z \cdot \frac{y}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   5. Taylor expanded in t around 0 55.0%

      \[\leadsto \frac{x + z \cdot \frac{y}{t}}{\color{blue}{\frac{b \cdot y + t \cdot \left(1 + a\right)}{t}}} \]

   6. Taylor expanded in x around 0 63.0%

      \[\leadsto \color{blue}{\frac{y \cdot z}{b \cdot y + t \cdot \left(1 + a\right)}} \]

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -5.00000000000000037e-286 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.0000000000000001e280

   1. Initial program 98.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   if -5.00000000000000037e-286 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0 or 4.0000000000000001e280 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 28.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. +-commutative 28.6%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 36.4%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. fma-define 36.4%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative 36.4%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]

      5. associate-/l* 42.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{y \cdot \frac{b}{t}} + \left(a + 1\right)} \]

      6. fma-define 42.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]

   3. Simplified 42.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 32.9%

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} + \frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 32.9%

         \[\leadsto \color{blue}{\frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]

      2. times-frac 36.2%

         \[\leadsto \color{blue}{\frac{y}{t} \cdot \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} \]

      3. fma-define 36.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right)} \]

      4. associate-*r/ 39.8%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      5. +-commutative 39.8%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      6. fma-define 39.8%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      7. associate-*r/ 43.7%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}\right) \]

      8. +-commutative 43.7%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}\right) \]

      9. fma-define 43.7%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}\right) \]

   7. Simplified 43.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\right)} \]

   8. Taylor expanded in b around inf 77.9%

      \[\leadsto \color{blue}{\frac{z + \frac{t \cdot x}{y}}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -5 \cdot 10^{-286} \lor \neg \left(\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0\right) \land \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 4 \cdot 10^{+280}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot b}{t}\\ t_2 := 1 + \left(a + t\_1\right)\\ t_3 := \frac{x + \frac{y \cdot z}{t}}{t\_1 + \left(a + 1\right)}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-286} \lor \neg \left(t\_3 \leq 0\right) \land t\_3 \leq 4 \cdot 10^{+280}:\\ \;\;\;\;\frac{x}{t\_2} + \frac{y \cdot z}{t \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* y b) t))
        (t_2 (+ 1.0 (+ a t_1)))
        (t_3 (/ (+ x (/ (* y z) t)) (+ t_1 (+ a 1.0)))))
   (if (or (<= t_3 -5e-286) (and (not (<= t_3 0.0)) (<= t_3 4e+280)))
     (+ (/ x t_2) (/ (* y z) (* t t_2)))
     (/ (+ z (/ (* x t) y)) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * b) / t;
	double t_2 = 1.0 + (a + t_1);
	double t_3 = (x + ((y * z) / t)) / (t_1 + (a + 1.0));
	double tmp;
	if ((t_3 <= -5e-286) || (!(t_3 <= 0.0) && (t_3 <= 4e+280))) {
		tmp = (x / t_2) + ((y * z) / (t * t_2));
	} else {
		tmp = (z + ((x * t) / y)) / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (y * b) / t
    t_2 = 1.0d0 + (a + t_1)
    t_3 = (x + ((y * z) / t)) / (t_1 + (a + 1.0d0))
    if ((t_3 <= (-5d-286)) .or. (.not. (t_3 <= 0.0d0)) .and. (t_3 <= 4d+280)) then
        tmp = (x / t_2) + ((y * z) / (t * t_2))
    else
        tmp = (z + ((x * t) / y)) / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * b) / t;
	double t_2 = 1.0 + (a + t_1);
	double t_3 = (x + ((y * z) / t)) / (t_1 + (a + 1.0));
	double tmp;
	if ((t_3 <= -5e-286) || (!(t_3 <= 0.0) && (t_3 <= 4e+280))) {
		tmp = (x / t_2) + ((y * z) / (t * t_2));
	} else {
		tmp = (z + ((x * t) / y)) / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (y * b) / t
	t_2 = 1.0 + (a + t_1)
	t_3 = (x + ((y * z) / t)) / (t_1 + (a + 1.0))
	tmp = 0
	if (t_3 <= -5e-286) or (not (t_3 <= 0.0) and (t_3 <= 4e+280)):
		tmp = (x / t_2) + ((y * z) / (t * t_2))
	else:
		tmp = (z + ((x * t) / y)) / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * b) / t)
	t_2 = Float64(1.0 + Float64(a + t_1))
	t_3 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(t_1 + Float64(a + 1.0)))
	tmp = 0.0
	if ((t_3 <= -5e-286) || (!(t_3 <= 0.0) && (t_3 <= 4e+280)))
		tmp = Float64(Float64(x / t_2) + Float64(Float64(y * z) / Float64(t * t_2)));
	else
		tmp = Float64(Float64(z + Float64(Float64(x * t) / y)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (y * b) / t;
	t_2 = 1.0 + (a + t_1);
	t_3 = (x + ((y * z) / t)) / (t_1 + (a + 1.0));
	tmp = 0.0;
	if ((t_3 <= -5e-286) || (~((t_3 <= 0.0)) && (t_3 <= 4e+280)))
		tmp = (x / t_2) + ((y * z) / (t * t_2));
	else
		tmp = (z + ((x * t) / y)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(a + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$3, -5e-286], And[N[Not[LessEqual[t$95$3, 0.0]], $MachinePrecision], LessEqual[t$95$3, 4e+280]]], N[(N[(x / t$95$2), $MachinePrecision] + N[(N[(y * z), $MachinePrecision] / N[(t * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(N[(x * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -5.00000000000000037e-286 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.0000000000000001e280

   1. Initial program 90.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. +-commutative 90.0%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 89.8%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. fma-define 89.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative 89.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]

      5. associate-/l* 85.6%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{y \cdot \frac{b}{t}} + \left(a + 1\right)} \]

      6. fma-define 85.6%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]

   3. Simplified 85.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 94.2%

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} + \frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}} \]

   if -5.00000000000000037e-286 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0 or 4.0000000000000001e280 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 28.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. +-commutative 28.6%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 36.4%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. fma-define 36.4%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative 36.4%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]

      5. associate-/l* 42.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{y \cdot \frac{b}{t}} + \left(a + 1\right)} \]

      6. fma-define 42.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]

   3. Simplified 42.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 32.9%

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} + \frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 32.9%

         \[\leadsto \color{blue}{\frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]

      2. times-frac 36.2%

         \[\leadsto \color{blue}{\frac{y}{t} \cdot \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} \]

      3. fma-define 36.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right)} \]

      4. associate-*r/ 39.8%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      5. +-commutative 39.8%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      6. fma-define 39.8%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      7. associate-*r/ 43.7%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}\right) \]

      8. +-commutative 43.7%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}\right) \]

      9. fma-define 43.7%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}\right) \]

   7. Simplified 43.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\right)} \]

   8. Taylor expanded in b around inf 77.9%

      \[\leadsto \color{blue}{\frac{z + \frac{t \cdot x}{y}}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -5 \cdot 10^{-286} \lor \neg \left(\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0\right) \land \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 4 \cdot 10^{+280}:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)} + \frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{y \cdot b}{t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{+207}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+184}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.55e+207)
   (/ (+ z (/ (* x t) y)) b)
   (if (<= y -2e-17)
     (/ (+ x (* y (/ z t))) (+ (+ a 1.0) (* y (/ b t))))
     (if (<= y 6.8e+184)
       (/ (+ x (* z (/ y t))) (+ (/ (* y b) t) (+ a 1.0)))
       (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.55e+207) {
		tmp = (z + ((x * t) / y)) / b;
	} else if (y <= -2e-17) {
		tmp = (x + (y * (z / t))) / ((a + 1.0) + (y * (b / t)));
	} else if (y <= 6.8e+184) {
		tmp = (x + (z * (y / t))) / (((y * b) / t) + (a + 1.0));
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-2.55d+207)) then
        tmp = (z + ((x * t) / y)) / b
    else if (y <= (-2d-17)) then
        tmp = (x + (y * (z / t))) / ((a + 1.0d0) + (y * (b / t)))
    else if (y <= 6.8d+184) then
        tmp = (x + (z * (y / t))) / (((y * b) / t) + (a + 1.0d0))
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.55e+207) {
		tmp = (z + ((x * t) / y)) / b;
	} else if (y <= -2e-17) {
		tmp = (x + (y * (z / t))) / ((a + 1.0) + (y * (b / t)));
	} else if (y <= 6.8e+184) {
		tmp = (x + (z * (y / t))) / (((y * b) / t) + (a + 1.0));
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.55e+207:
		tmp = (z + ((x * t) / y)) / b
	elif y <= -2e-17:
		tmp = (x + (y * (z / t))) / ((a + 1.0) + (y * (b / t)))
	elif y <= 6.8e+184:
		tmp = (x + (z * (y / t))) / (((y * b) / t) + (a + 1.0))
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.55e+207)
		tmp = Float64(Float64(z + Float64(Float64(x * t) / y)) / b);
	elseif (y <= -2e-17)
		tmp = Float64(Float64(x + Float64(y * Float64(z / t))) / Float64(Float64(a + 1.0) + Float64(y * Float64(b / t))));
	elseif (y <= 6.8e+184)
		tmp = Float64(Float64(x + Float64(z * Float64(y / t))) / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2.55e+207)
		tmp = (z + ((x * t) / y)) / b;
	elseif (y <= -2e-17)
		tmp = (x + (y * (z / t))) / ((a + 1.0) + (y * (b / t)));
	elseif (y <= 6.8e+184)
		tmp = (x + (z * (y / t))) / (((y * b) / t) + (a + 1.0));
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.55e+207], N[(N[(z + N[(N[(x * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[y, -2e-17], N[(N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(y * N[(b / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.8e+184], N[(N[(x + N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.5499999999999999e207

   1. Initial program 24.9%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. +-commutative 24.9%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 38.5%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. fma-define 38.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative 38.5%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]

      5. associate-/l* 47.0%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{y \cdot \frac{b}{t}} + \left(a + 1\right)} \]

      6. fma-define 47.0%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]

   3. Simplified 47.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 25.0%

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} + \frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 25.0%

         \[\leadsto \color{blue}{\frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]

      2. times-frac 29.5%

         \[\leadsto \color{blue}{\frac{y}{t} \cdot \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} \]

      3. fma-define 29.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right)} \]

      4. associate-*r/ 28.3%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      5. +-commutative 28.3%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      6. fma-define 28.3%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      7. associate-*r/ 28.3%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}\right) \]

      8. +-commutative 28.3%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}\right) \]

      9. fma-define 28.3%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}\right) \]

   7. Simplified 28.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\right)} \]

   8. Taylor expanded in b around inf 77.9%

      \[\leadsto \color{blue}{\frac{z + \frac{t \cdot x}{y}}{b}} \]

   if -2.5499999999999999e207 < y < -2.00000000000000014e-17

   1. Initial program 67.2%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. associate-/l* 71.4%

         \[\leadsto \frac{x + \color{blue}{y \cdot \frac{z}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 77.4%

         \[\leadsto \frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \color{blue}{y \cdot \frac{b}{t}}} \]

   3. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}} \]
   4. Add Preprocessing

   if -2.00000000000000014e-17 < y < 6.8000000000000003e184

   1. Initial program 86.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 86.8%

         \[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 89.4%

         \[\leadsto \frac{x + \color{blue}{z \cdot \frac{y}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   4. Applied egg-rr 89.4%

      \[\leadsto \frac{x + \color{blue}{z \cdot \frac{y}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   if 6.8000000000000003e184 < y

   1. Initial program 34.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. +-commutative 34.1%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 40.7%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. fma-define 40.7%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative 40.7%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]

      5. associate-/l* 41.0%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{y \cdot \frac{b}{t}} + \left(a + 1\right)} \]

      6. fma-define 41.0%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]

   3. Simplified 41.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 79.0%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{+207}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+184}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-73} \lor \neg \left(t \leq 2.5 \cdot 10^{-200}\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -5.2e-73) (not (<= t 2.5e-200)))
   (/ (+ x (* y (/ z t))) (+ (+ a 1.0) (* y (/ b t))))
   (/ (+ z (/ (* x t) y)) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -5.2e-73) || !(t <= 2.5e-200)) {
		tmp = (x + (y * (z / t))) / ((a + 1.0) + (y * (b / t)));
	} else {
		tmp = (z + ((x * t) / y)) / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-5.2d-73)) .or. (.not. (t <= 2.5d-200))) then
        tmp = (x + (y * (z / t))) / ((a + 1.0d0) + (y * (b / t)))
    else
        tmp = (z + ((x * t) / y)) / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -5.2e-73) || !(t <= 2.5e-200)) {
		tmp = (x + (y * (z / t))) / ((a + 1.0) + (y * (b / t)));
	} else {
		tmp = (z + ((x * t) / y)) / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -5.2e-73) or not (t <= 2.5e-200):
		tmp = (x + (y * (z / t))) / ((a + 1.0) + (y * (b / t)))
	else:
		tmp = (z + ((x * t) / y)) / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -5.2e-73) || !(t <= 2.5e-200))
		tmp = Float64(Float64(x + Float64(y * Float64(z / t))) / Float64(Float64(a + 1.0) + Float64(y * Float64(b / t))));
	else
		tmp = Float64(Float64(z + Float64(Float64(x * t) / y)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -5.2e-73) || ~((t <= 2.5e-200)))
		tmp = (x + (y * (z / t))) / ((a + 1.0) + (y * (b / t)));
	else
		tmp = (z + ((x * t) / y)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -5.2e-73], N[Not[LessEqual[t, 2.5e-200]], $MachinePrecision]], N[(N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(y * N[(b / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(N[(x * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.2000000000000002e-73 or 2.49999999999999996e-200 < t

   1. Initial program 79.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. associate-/l* 83.8%

         \[\leadsto \frac{x + \color{blue}{y \cdot \frac{z}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 84.8%

         \[\leadsto \frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \color{blue}{y \cdot \frac{b}{t}}} \]

   3. Simplified 84.8%

      \[\leadsto \color{blue}{\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}} \]
   4. Add Preprocessing

   if -5.2000000000000002e-73 < t < 2.49999999999999996e-200

   1. Initial program 51.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. +-commutative 51.1%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 45.9%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. fma-define 46.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative 46.0%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]

      5. associate-/l* 38.6%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{y \cdot \frac{b}{t}} + \left(a + 1\right)} \]

      6. fma-define 38.6%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]

   3. Simplified 38.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 62.0%

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} + \frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 62.0%

         \[\leadsto \color{blue}{\frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]

      2. times-frac 51.3%

         \[\leadsto \color{blue}{\frac{y}{t} \cdot \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} \]

      3. fma-define 51.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right)} \]

      4. associate-*r/ 46.4%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      5. +-commutative 46.4%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      6. fma-define 46.4%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      7. associate-*r/ 46.4%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}\right) \]

      8. +-commutative 46.4%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}\right) \]

      9. fma-define 46.4%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}\right) \]

   7. Simplified 46.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\right)} \]

   8. Taylor expanded in b around inf 76.7%

      \[\leadsto \color{blue}{\frac{z + \frac{t \cdot x}{y}}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-73} \lor \neg \left(t \leq 2.5 \cdot 10^{-200}\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -500000000000 \lor \neg \left(t \leq 4.4 \cdot 10^{-98}\right):\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -500000000000.0) (not (<= t 4.4e-98)))
   (/ (+ x (* z (/ y t))) (+ a 1.0))
   (/ (+ z (/ (* x t) y)) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -500000000000.0) || !(t <= 4.4e-98)) {
		tmp = (x + (z * (y / t))) / (a + 1.0);
	} else {
		tmp = (z + ((x * t) / y)) / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-500000000000.0d0)) .or. (.not. (t <= 4.4d-98))) then
        tmp = (x + (z * (y / t))) / (a + 1.0d0)
    else
        tmp = (z + ((x * t) / y)) / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -500000000000.0) || !(t <= 4.4e-98)) {
		tmp = (x + (z * (y / t))) / (a + 1.0);
	} else {
		tmp = (z + ((x * t) / y)) / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -500000000000.0) or not (t <= 4.4e-98):
		tmp = (x + (z * (y / t))) / (a + 1.0)
	else:
		tmp = (z + ((x * t) / y)) / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -500000000000.0) || !(t <= 4.4e-98))
		tmp = Float64(Float64(x + Float64(z * Float64(y / t))) / Float64(a + 1.0));
	else
		tmp = Float64(Float64(z + Float64(Float64(x * t) / y)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -500000000000.0) || ~((t <= 4.4e-98)))
		tmp = (x + (z * (y / t))) / (a + 1.0);
	else
		tmp = (z + ((x * t) / y)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -500000000000.0], N[Not[LessEqual[t, 4.4e-98]], $MachinePrecision]], N[(N[(x + N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(N[(x * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5e11 or 4.39999999999999993e-98 < t

   1. Initial program 80.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 80.1%

         \[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 87.4%

         \[\leadsto \frac{x + \color{blue}{z \cdot \frac{y}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   4. Applied egg-rr 87.4%

      \[\leadsto \frac{x + \color{blue}{z \cdot \frac{y}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   5. Taylor expanded in y around 0 78.7%

      \[\leadsto \frac{x + z \cdot \frac{y}{t}}{\color{blue}{1 + a}} \]

   if -5e11 < t < 4.39999999999999993e-98

   1. Initial program 61.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. +-commutative 61.1%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 57.1%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. fma-define 57.2%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative 57.2%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]

      5. associate-/l* 51.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{y \cdot \frac{b}{t}} + \left(a + 1\right)} \]

      6. fma-define 51.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]

   3. Simplified 51.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 70.2%

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} + \frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 70.2%

         \[\leadsto \color{blue}{\frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]

      2. times-frac 58.5%

         \[\leadsto \color{blue}{\frac{y}{t} \cdot \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} \]

      3. fma-define 58.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right)} \]

      4. associate-*r/ 53.6%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      5. +-commutative 53.6%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      6. fma-define 53.6%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      7. associate-*r/ 53.6%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}\right) \]

      8. +-commutative 53.6%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}\right) \]

      9. fma-define 53.6%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}\right) \]

   7. Simplified 53.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\right)} \]

   8. Taylor expanded in b around inf 69.1%

      \[\leadsto \color{blue}{\frac{z + \frac{t \cdot x}{y}}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -500000000000 \lor \neg \left(t \leq 4.4 \cdot 10^{-98}\right):\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -31000000000000 \lor \neg \left(t \leq 8.5 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -31000000000000.0) (not (<= t 8.5e-56)))
   (/ x (+ 1.0 (+ a (/ (* y b) t))))
   (/ (+ z (/ (* x t) y)) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -31000000000000.0) || !(t <= 8.5e-56)) {
		tmp = x / (1.0 + (a + ((y * b) / t)));
	} else {
		tmp = (z + ((x * t) / y)) / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-31000000000000.0d0)) .or. (.not. (t <= 8.5d-56))) then
        tmp = x / (1.0d0 + (a + ((y * b) / t)))
    else
        tmp = (z + ((x * t) / y)) / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -31000000000000.0) || !(t <= 8.5e-56)) {
		tmp = x / (1.0 + (a + ((y * b) / t)));
	} else {
		tmp = (z + ((x * t) / y)) / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -31000000000000.0) or not (t <= 8.5e-56):
		tmp = x / (1.0 + (a + ((y * b) / t)))
	else:
		tmp = (z + ((x * t) / y)) / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -31000000000000.0) || !(t <= 8.5e-56))
		tmp = Float64(x / Float64(1.0 + Float64(a + Float64(Float64(y * b) / t))));
	else
		tmp = Float64(Float64(z + Float64(Float64(x * t) / y)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -31000000000000.0) || ~((t <= 8.5e-56)))
		tmp = x / (1.0 + (a + ((y * b) / t)));
	else
		tmp = (z + ((x * t) / y)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -31000000000000.0], N[Not[LessEqual[t, 8.5e-56]], $MachinePrecision]], N[(x / N[(1.0 + N[(a + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(N[(x * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.1e13 or 8.49999999999999932e-56 < t

   1. Initial program 80.5%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. +-commutative 80.5%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 88.4%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. fma-define 88.4%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative 88.4%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]

      5. associate-/l* 92.0%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{y \cdot \frac{b}{t}} + \left(a + 1\right)} \]

      6. fma-define 92.0%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]

   3. Simplified 92.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 69.0%

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]

   if -3.1e13 < t < 8.49999999999999932e-56

   1. Initial program 62.5%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. +-commutative 62.5%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 58.2%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. fma-define 58.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative 58.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]

      5. associate-/l* 51.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{y \cdot \frac{b}{t}} + \left(a + 1\right)} \]

      6. fma-define 51.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]

   3. Simplified 51.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 71.5%

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} + \frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 71.5%

         \[\leadsto \color{blue}{\frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]

      2. times-frac 61.8%

         \[\leadsto \color{blue}{\frac{y}{t} \cdot \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} \]

      3. fma-define 61.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right)} \]

      4. associate-*r/ 56.5%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      5. +-commutative 56.5%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      6. fma-define 56.5%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      7. associate-*r/ 56.5%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}\right) \]

      8. +-commutative 56.5%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}\right) \]

      9. fma-define 56.5%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}\right) \]

   7. Simplified 56.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\right)} \]

   8. Taylor expanded in b around inf 66.6%

      \[\leadsto \color{blue}{\frac{z + \frac{t \cdot x}{y}}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -31000000000000 \lor \neg \left(t \leq 8.5 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9000000000:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-47}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a + 1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -9000000000.0)
   (/ x (+ a 1.0))
   (if (<= t 1.75e-47) (/ (+ z (/ (* x t) y)) b) (/ 1.0 (/ (+ a 1.0) x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -9000000000.0) {
		tmp = x / (a + 1.0);
	} else if (t <= 1.75e-47) {
		tmp = (z + ((x * t) / y)) / b;
	} else {
		tmp = 1.0 / ((a + 1.0) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= (-9000000000.0d0)) then
        tmp = x / (a + 1.0d0)
    else if (t <= 1.75d-47) then
        tmp = (z + ((x * t) / y)) / b
    else
        tmp = 1.0d0 / ((a + 1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -9000000000.0) {
		tmp = x / (a + 1.0);
	} else if (t <= 1.75e-47) {
		tmp = (z + ((x * t) / y)) / b;
	} else {
		tmp = 1.0 / ((a + 1.0) / x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -9000000000.0:
		tmp = x / (a + 1.0)
	elif t <= 1.75e-47:
		tmp = (z + ((x * t) / y)) / b
	else:
		tmp = 1.0 / ((a + 1.0) / x)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -9000000000.0)
		tmp = Float64(x / Float64(a + 1.0));
	elseif (t <= 1.75e-47)
		tmp = Float64(Float64(z + Float64(Float64(x * t) / y)) / b);
	else
		tmp = Float64(1.0 / Float64(Float64(a + 1.0) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -9000000000.0)
		tmp = x / (a + 1.0);
	elseif (t <= 1.75e-47)
		tmp = (z + ((x * t) / y)) / b;
	else
		tmp = 1.0 / ((a + 1.0) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -9000000000.0], N[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.75e-47], N[(N[(z + N[(N[(x * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], N[(1.0 / N[(N[(a + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9e9

   1. Initial program 80.5%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. +-commutative 80.5%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 85.7%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. fma-define 85.7%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative 85.7%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]

      5. associate-/l* 92.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{y \cdot \frac{b}{t}} + \left(a + 1\right)} \]

      6. fma-define 92.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]

   3. Simplified 92.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 62.3%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   if -9e9 < t < 1.7499999999999999e-47

   1. Initial program 63.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. +-commutative 63.1%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 58.9%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. fma-define 58.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative 58.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]

      5. associate-/l* 52.7%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{y \cdot \frac{b}{t}} + \left(a + 1\right)} \]

      6. fma-define 52.7%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]

   3. Simplified 52.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 72.0%

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} + \frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 72.0%

         \[\leadsto \color{blue}{\frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]

      2. times-frac 62.4%

         \[\leadsto \color{blue}{\frac{y}{t} \cdot \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} \]

      3. fma-define 62.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right)} \]

      4. associate-*r/ 57.2%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      5. +-commutative 57.2%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      6. fma-define 57.2%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      7. associate-*r/ 57.2%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}\right) \]

      8. +-commutative 57.2%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}\right) \]

      9. fma-define 57.2%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}\right) \]

   7. Simplified 57.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\right)} \]

   8. Taylor expanded in b around inf 66.4%

      \[\leadsto \color{blue}{\frac{z + \frac{t \cdot x}{y}}{b}} \]

   if 1.7499999999999999e-47 < t

   1. Initial program 80.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 80.1%

         \[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 90.0%

         \[\leadsto \frac{x + \color{blue}{z \cdot \frac{y}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   4. Applied egg-rr 90.0%

      \[\leadsto \frac{x + \color{blue}{z \cdot \frac{y}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   5. Step-by-step derivation

      1. clear-num 89.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(a + 1\right) + \frac{y \cdot b}{t}}{x + z \cdot \frac{y}{t}}}} \]

      2. inv-pow 89.8%

         \[\leadsto \color{blue}{{\left(\frac{\left(a + 1\right) + \frac{y \cdot b}{t}}{x + z \cdot \frac{y}{t}}\right)}^{-1}} \]

      3. +-commutative 89.8%

         \[\leadsto {\left(\frac{\color{blue}{\left(1 + a\right)} + \frac{y \cdot b}{t}}{x + z \cdot \frac{y}{t}}\right)}^{-1} \]

      4. *-commutative 89.8%

         \[\leadsto {\left(\frac{\left(1 + a\right) + \frac{\color{blue}{b \cdot y}}{t}}{x + z \cdot \frac{y}{t}}\right)}^{-1} \]

      5. associate-*r/ 92.4%

         \[\leadsto {\left(\frac{\left(1 + a\right) + \color{blue}{b \cdot \frac{y}{t}}}{x + z \cdot \frac{y}{t}}\right)}^{-1} \]

      6. associate-+r+ 92.4%

         \[\leadsto {\left(\frac{\color{blue}{1 + \left(a + b \cdot \frac{y}{t}\right)}}{x + z \cdot \frac{y}{t}}\right)}^{-1} \]

      7. +-commutative 92.4%

         \[\leadsto {\left(\frac{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}{x + z \cdot \frac{y}{t}}\right)}^{-1} \]

      8. associate-*r/ 89.8%

         \[\leadsto {\left(\frac{1 + \left(\color{blue}{\frac{b \cdot y}{t}} + a\right)}{x + z \cdot \frac{y}{t}}\right)}^{-1} \]

      9. *-commutative 89.8%

         \[\leadsto {\left(\frac{1 + \left(\frac{\color{blue}{y \cdot b}}{t} + a\right)}{x + z \cdot \frac{y}{t}}\right)}^{-1} \]

      10. associate-/l* 91.1%

          \[\leadsto {\left(\frac{1 + \left(\color{blue}{y \cdot \frac{b}{t}} + a\right)}{x + z \cdot \frac{y}{t}}\right)}^{-1} \]

      11. fma-define 91.1%

          \[\leadsto {\left(\frac{1 + \color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a\right)}}{x + z \cdot \frac{y}{t}}\right)}^{-1} \]

      12. +-commutative 91.1%

          \[\leadsto {\left(\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{z \cdot \frac{y}{t} + x}}\right)}^{-1} \]

      13. fma-define 91.1%

          \[\leadsto {\left(\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}\right)}^{-1} \]

   6. Applied egg-rr 91.1%

      \[\leadsto \color{blue}{{\left(\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 91.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}} \]

   8. Simplified 91.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}} \]

   9. Taylor expanded in y around 0 65.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + a}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9000000000:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-47}:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a + 1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -29000000000000:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-49}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a + 1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -29000000000000.0)
   (/ x (+ a 1.0))
   (if (<= t 4e-49) (/ z b) (/ 1.0 (/ (+ a 1.0) x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -29000000000000.0) {
		tmp = x / (a + 1.0);
	} else if (t <= 4e-49) {
		tmp = z / b;
	} else {
		tmp = 1.0 / ((a + 1.0) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= (-29000000000000.0d0)) then
        tmp = x / (a + 1.0d0)
    else if (t <= 4d-49) then
        tmp = z / b
    else
        tmp = 1.0d0 / ((a + 1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -29000000000000.0) {
		tmp = x / (a + 1.0);
	} else if (t <= 4e-49) {
		tmp = z / b;
	} else {
		tmp = 1.0 / ((a + 1.0) / x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -29000000000000.0:
		tmp = x / (a + 1.0)
	elif t <= 4e-49:
		tmp = z / b
	else:
		tmp = 1.0 / ((a + 1.0) / x)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -29000000000000.0)
		tmp = Float64(x / Float64(a + 1.0));
	elseif (t <= 4e-49)
		tmp = Float64(z / b);
	else
		tmp = Float64(1.0 / Float64(Float64(a + 1.0) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -29000000000000.0)
		tmp = x / (a + 1.0);
	elseif (t <= 4e-49)
		tmp = z / b;
	else
		tmp = 1.0 / ((a + 1.0) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -29000000000000.0], N[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4e-49], N[(z / b), $MachinePrecision], N[(1.0 / N[(N[(a + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.9e13

   1. Initial program 80.5%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. +-commutative 80.5%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 85.7%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. fma-define 85.7%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative 85.7%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]

      5. associate-/l* 92.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{y \cdot \frac{b}{t}} + \left(a + 1\right)} \]

      6. fma-define 92.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]

   3. Simplified 92.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 62.3%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   if -2.9e13 < t < 3.99999999999999975e-49

   1. Initial program 63.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. +-commutative 63.1%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 58.9%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. fma-define 58.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative 58.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]

      5. associate-/l* 52.7%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{y \cdot \frac{b}{t}} + \left(a + 1\right)} \]

      6. fma-define 52.7%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]

   3. Simplified 52.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 57.4%

      \[\leadsto \color{blue}{\frac{z}{b}} \]

   if 3.99999999999999975e-49 < t

   1. Initial program 80.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 80.1%

         \[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 90.0%

         \[\leadsto \frac{x + \color{blue}{z \cdot \frac{y}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   4. Applied egg-rr 90.0%

      \[\leadsto \frac{x + \color{blue}{z \cdot \frac{y}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   5. Step-by-step derivation

      1. clear-num 89.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(a + 1\right) + \frac{y \cdot b}{t}}{x + z \cdot \frac{y}{t}}}} \]

      2. inv-pow 89.8%

         \[\leadsto \color{blue}{{\left(\frac{\left(a + 1\right) + \frac{y \cdot b}{t}}{x + z \cdot \frac{y}{t}}\right)}^{-1}} \]

      3. +-commutative 89.8%

         \[\leadsto {\left(\frac{\color{blue}{\left(1 + a\right)} + \frac{y \cdot b}{t}}{x + z \cdot \frac{y}{t}}\right)}^{-1} \]

      4. *-commutative 89.8%

         \[\leadsto {\left(\frac{\left(1 + a\right) + \frac{\color{blue}{b \cdot y}}{t}}{x + z \cdot \frac{y}{t}}\right)}^{-1} \]

      5. associate-*r/ 92.4%

         \[\leadsto {\left(\frac{\left(1 + a\right) + \color{blue}{b \cdot \frac{y}{t}}}{x + z \cdot \frac{y}{t}}\right)}^{-1} \]

      6. associate-+r+ 92.4%

         \[\leadsto {\left(\frac{\color{blue}{1 + \left(a + b \cdot \frac{y}{t}\right)}}{x + z \cdot \frac{y}{t}}\right)}^{-1} \]

      7. +-commutative 92.4%

         \[\leadsto {\left(\frac{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}{x + z \cdot \frac{y}{t}}\right)}^{-1} \]

      8. associate-*r/ 89.8%

         \[\leadsto {\left(\frac{1 + \left(\color{blue}{\frac{b \cdot y}{t}} + a\right)}{x + z \cdot \frac{y}{t}}\right)}^{-1} \]

      9. *-commutative 89.8%

         \[\leadsto {\left(\frac{1 + \left(\frac{\color{blue}{y \cdot b}}{t} + a\right)}{x + z \cdot \frac{y}{t}}\right)}^{-1} \]

      10. associate-/l* 91.1%

          \[\leadsto {\left(\frac{1 + \left(\color{blue}{y \cdot \frac{b}{t}} + a\right)}{x + z \cdot \frac{y}{t}}\right)}^{-1} \]

      11. fma-define 91.1%

          \[\leadsto {\left(\frac{1 + \color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a\right)}}{x + z \cdot \frac{y}{t}}\right)}^{-1} \]

      12. +-commutative 91.1%

          \[\leadsto {\left(\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{z \cdot \frac{y}{t} + x}}\right)}^{-1} \]

      13. fma-define 91.1%

          \[\leadsto {\left(\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}\right)}^{-1} \]

   6. Applied egg-rr 91.1%

      \[\leadsto \color{blue}{{\left(\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 91.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}} \]

   8. Simplified 91.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}} \]

   9. Taylor expanded in y around 0 65.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + a}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -29000000000000:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-49}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a + 1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -210000000000 \lor \neg \left(t \leq 9 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -210000000000.0) (not (<= t 9e-49))) (/ x (+ a 1.0)) (/ z b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -210000000000.0) || !(t <= 9e-49)) {
		tmp = x / (a + 1.0);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-210000000000.0d0)) .or. (.not. (t <= 9d-49))) then
        tmp = x / (a + 1.0d0)
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -210000000000.0) || !(t <= 9e-49)) {
		tmp = x / (a + 1.0);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -210000000000.0) or not (t <= 9e-49):
		tmp = x / (a + 1.0)
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -210000000000.0) || !(t <= 9e-49))
		tmp = Float64(x / Float64(a + 1.0));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -210000000000.0) || ~((t <= 9e-49)))
		tmp = x / (a + 1.0);
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -210000000000.0], N[Not[LessEqual[t, 9e-49]], $MachinePrecision]], N[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.1e11 or 9.0000000000000004e-49 < t

   1. Initial program 80.2%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. +-commutative 80.2%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 88.3%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. fma-define 88.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative 88.3%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]

      5. associate-/l* 91.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{y \cdot \frac{b}{t}} + \left(a + 1\right)} \]

      6. fma-define 91.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]

   3. Simplified 91.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 64.0%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   if -2.1e11 < t < 9.0000000000000004e-49

   1. Initial program 63.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. +-commutative 63.1%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 58.9%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. fma-define 58.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative 58.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]

      5. associate-/l* 52.7%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{y \cdot \frac{b}{t}} + \left(a + 1\right)} \]

      6. fma-define 52.7%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]

   3. Simplified 52.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 57.4%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -210000000000 \lor \neg \left(t \leq 9 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -48000000000 \lor \neg \left(t \leq 2.9 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -48000000000.0) (not (<= t 2.9e+23))) (/ x a) (/ z b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -48000000000.0) || !(t <= 2.9e+23)) {
		tmp = x / a;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-48000000000.0d0)) .or. (.not. (t <= 2.9d+23))) then
        tmp = x / a
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -48000000000.0) || !(t <= 2.9e+23)) {
		tmp = x / a;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -48000000000.0) or not (t <= 2.9e+23):
		tmp = x / a
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -48000000000.0) || !(t <= 2.9e+23))
		tmp = Float64(x / a);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -48000000000.0) || ~((t <= 2.9e+23)))
		tmp = x / a;
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -48000000000.0], N[Not[LessEqual[t, 2.9e+23]], $MachinePrecision]], N[(x / a), $MachinePrecision], N[(z / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.8e10 or 2.90000000000000013e23 < t

   1. Initial program 79.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. +-commutative 79.7%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 88.6%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. fma-define 88.6%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative 88.6%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]

      5. associate-/l* 93.4%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{y \cdot \frac{b}{t}} + \left(a + 1\right)} \]

      6. fma-define 93.4%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]

   3. Simplified 93.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 79.7%

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} + \frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 79.7%

         \[\leadsto \color{blue}{\frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]

      2. times-frac 89.5%

         \[\leadsto \color{blue}{\frac{y}{t} \cdot \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} \]

      3. fma-define 89.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right)} \]

      4. associate-*r/ 92.7%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      5. +-commutative 92.7%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      6. fma-define 92.7%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

      7. associate-*r/ 95.1%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}\right) \]

      8. +-commutative 95.1%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}\right) \]

      9. fma-define 95.1%

         \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}\right) \]

   7. Simplified 95.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\right)} \]

   8. Taylor expanded in z around inf 76.8%

      \[\leadsto \color{blue}{z \cdot \left(\frac{x}{z \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)} + \frac{y}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}\right)} \]
   9. Step-by-step derivation

      1. associate-/r* 74.2%

         \[\leadsto z \cdot \left(\color{blue}{\frac{\frac{x}{z}}{1 + \left(a + \frac{b \cdot y}{t}\right)}} + \frac{y}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}\right) \]

      2. associate-/l* 75.9%

         \[\leadsto z \cdot \left(\frac{\frac{x}{z}}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)} + \frac{y}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}\right) \]

      3. associate-/r* 76.7%

         \[\leadsto z \cdot \left(\frac{\frac{x}{z}}{1 + \left(a + b \cdot \frac{y}{t}\right)} + \color{blue}{\frac{\frac{y}{t}}{1 + \left(a + \frac{b \cdot y}{t}\right)}}\right) \]

      4. associate-/l* 80.7%

         \[\leadsto z \cdot \left(\frac{\frac{x}{z}}{1 + \left(a + b \cdot \frac{y}{t}\right)} + \frac{\frac{y}{t}}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}\right) \]

   10. Simplified 80.7%

       \[\leadsto \color{blue}{z \cdot \left(\frac{\frac{x}{z}}{1 + \left(a + b \cdot \frac{y}{t}\right)} + \frac{\frac{y}{t}}{1 + \left(a + b \cdot \frac{y}{t}\right)}\right)} \]

   11. Taylor expanded in a around inf 35.3%

       \[\leadsto z \cdot \color{blue}{\frac{\frac{x}{z} + \frac{y}{t}}{a}} \]

   12. Taylor expanded in z around 0 35.5%

       \[\leadsto \color{blue}{\frac{x}{a}} \]

   if -4.8e10 < t < 2.90000000000000013e23

   1. Initial program 65.2%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Step-by-step derivation

      1. +-commutative 65.2%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      2. associate-/l* 61.4%

         \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      3. fma-define 61.4%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

      4. +-commutative 61.4%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]

      5. associate-/l* 55.1%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{y \cdot \frac{b}{t}} + \left(a + 1\right)} \]

      6. fma-define 55.1%

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]

   3. Simplified 55.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 55.2%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -48000000000 \lor \neg \left(t \leq 2.9 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{a} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ x a))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x / a;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x / a
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x / a;
}

Python

def code(x, y, z, t, a, b):
	return x / a

Julia

function code(x, y, z, t, a, b)
	return Float64(x / a)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x / a;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x / a), $MachinePrecision]

Derivation

1. Initial program 72.0%

   \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
2. Step-by-step derivation

   1. +-commutative 72.0%

      \[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   2. associate-/l* 74.2%

      \[\leadsto \frac{\color{blue}{y \cdot \frac{z}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   3. fma-define 74.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   4. +-commutative 74.2%

      \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\frac{y \cdot b}{t} + \left(a + 1\right)}} \]

   5. associate-/l* 73.1%

      \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{y \cdot \frac{b}{t}} + \left(a + 1\right)} \]

   6. fma-define 73.1%

      \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]

3. Simplified 73.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}} \]
4. Add Preprocessing

5. Taylor expanded in z around 0 76.3%

   \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} + \frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}} \]
6. Step-by-step derivation

   1. +-commutative 76.3%

      \[\leadsto \color{blue}{\frac{y \cdot z}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]

   2. times-frac 76.3%

      \[\leadsto \color{blue}{\frac{y}{t} \cdot \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}} + \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)} \]

   3. fma-define 76.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \frac{b \cdot y}{t}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right)} \]

   4. associate-*r/ 75.3%

      \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

   5. +-commutative 75.3%

      \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

   6. fma-define 75.3%

      \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}, \frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}\right) \]

   7. associate-*r/ 76.4%

      \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}\right) \]

   8. +-commutative 76.4%

      \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\left(b \cdot \frac{y}{t} + a\right)}}\right) \]

   9. fma-define 76.4%

      \[\leadsto \mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, a\right)}}\right) \]

7. Simplified 76.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}, \frac{x}{1 + \mathsf{fma}\left(b, \frac{y}{t}, a\right)}\right)} \]

8. Taylor expanded in z around inf 75.5%

   \[\leadsto \color{blue}{z \cdot \left(\frac{x}{z \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)} + \frac{y}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}\right)} \]
9. Step-by-step derivation

   1. associate-/r* 72.8%

      \[\leadsto z \cdot \left(\color{blue}{\frac{\frac{x}{z}}{1 + \left(a + \frac{b \cdot y}{t}\right)}} + \frac{y}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}\right) \]

   2. associate-/l* 73.5%

      \[\leadsto z \cdot \left(\frac{\frac{x}{z}}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)} + \frac{y}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}\right) \]

   3. associate-/r* 68.0%

      \[\leadsto z \cdot \left(\frac{\frac{x}{z}}{1 + \left(a + b \cdot \frac{y}{t}\right)} + \color{blue}{\frac{\frac{y}{t}}{1 + \left(a + \frac{b \cdot y}{t}\right)}}\right) \]

   4. associate-/l* 67.3%

      \[\leadsto z \cdot \left(\frac{\frac{x}{z}}{1 + \left(a + b \cdot \frac{y}{t}\right)} + \frac{\frac{y}{t}}{1 + \left(a + \color{blue}{b \cdot \frac{y}{t}}\right)}\right) \]

10. Simplified 67.3%

    \[\leadsto \color{blue}{z \cdot \left(\frac{\frac{x}{z}}{1 + \left(a + b \cdot \frac{y}{t}\right)} + \frac{\frac{y}{t}}{1 + \left(a + b \cdot \frac{y}{t}\right)}\right)} \]

11. Taylor expanded in a around inf 26.0%

    \[\leadsto z \cdot \color{blue}{\frac{\frac{x}{z} + \frac{y}{t}}{a}} \]

12. Taylor expanded in z around 0 22.8%

    \[\leadsto \color{blue}{\frac{x}{a}} \]
13. Add Preprocessing

Developer Target 1: 79.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{if}\;t < -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))
   (if (< t -1.3659085366310088e-271)
     t_1
     (if (< t 3.036967103737246e-130) (/ z b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	double tmp;
	if (t < -1.3659085366310088e-271) {
		tmp = t_1;
	} else if (t < 3.036967103737246e-130) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 * ((x + ((y / t) * z)) * (1.0d0 / ((a + 1.0d0) + ((y / t) * b))))
    if (t < (-1.3659085366310088d-271)) then
        tmp = t_1
    else if (t < 3.036967103737246d-130) then
        tmp = z / b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	double tmp;
	if (t < -1.3659085366310088e-271) {
		tmp = t_1;
	} else if (t < 3.036967103737246e-130) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))))
	tmp = 0
	if t < -1.3659085366310088e-271:
		tmp = t_1
	elif t < 3.036967103737246e-130:
		tmp = z / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 * Float64(Float64(x + Float64(Float64(y / t) * z)) * Float64(1.0 / Float64(Float64(a + 1.0) + Float64(Float64(y / t) * b)))))
	tmp = 0.0
	if (t < -1.3659085366310088e-271)
		tmp = t_1;
	elseif (t < 3.036967103737246e-130)
		tmp = Float64(z / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	tmp = 0.0;
	if (t < -1.3659085366310088e-271)
		tmp = t_1;
	elseif (t < 3.036967103737246e-130)
		tmp = z / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 * N[(N[(x + N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y / t), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -1.3659085366310088e-271], t$95$1, If[Less[t, 3.036967103737246e-130], N[(z / b), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024167 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -1707385670788761/12500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))) (if (< t 1518483551868623/5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))