Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 67.3% → 96.4%

Time: 13.9s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

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))) / (y + (z * (b - y)))
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))) / (y + (z * (b - y)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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))) / (y + (z * (b - y)))
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))) / (y + (z * (b - y)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b} - \frac{\mathsf{fma}\left(-y, \frac{x}{b - y}, \frac{y}{{\left(b - y\right)}^{2}} \cdot \left(t - a\right)\right)}{z}\\ t_2 := \frac{y \cdot x - \left(a - t\right) \cdot z}{\left(b - y\right) \cdot z + y}\\ t_3 := \mathsf{fma}\left(b - y, z, y\right)\\ t_4 := \mathsf{fma}\left(\frac{y}{t\_3}, x, \frac{z}{t\_3} \cdot \left(t - a\right)\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-184}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (-
          (/ (- a t) (- y b))
          (/ (fma (- y) (/ x (- b y)) (* (/ y (pow (- b y) 2.0)) (- t a))) z)))
        (t_2 (/ (- (* y x) (* (- a t) z)) (+ (* (- b y) z) y)))
        (t_3 (fma (- b y) z y))
        (t_4 (fma (/ y t_3) x (* (/ z t_3) (- t a)))))
   (if (<= t_2 -2e-184)
     t_4
     (if (<= t_2 0.0) t_1 (if (<= t_2 INFINITY) t_4 t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((a - t) / (y - b)) - (fma(-y, (x / (b - y)), ((y / pow((b - y), 2.0)) * (t - a))) / z);
	double t_2 = ((y * x) - ((a - t) * z)) / (((b - y) * z) + y);
	double t_3 = fma((b - y), z, y);
	double t_4 = fma((y / t_3), x, ((z / t_3) * (t - a)));
	double tmp;
	if (t_2 <= -2e-184) {
		tmp = t_4;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(a - t) / Float64(y - b)) - Float64(fma(Float64(-y), Float64(x / Float64(b - y)), Float64(Float64(y / (Float64(b - y) ^ 2.0)) * Float64(t - a))) / z))
	t_2 = Float64(Float64(Float64(y * x) - Float64(Float64(a - t) * z)) / Float64(Float64(Float64(b - y) * z) + y))
	t_3 = fma(Float64(b - y), z, y)
	t_4 = fma(Float64(y / t_3), x, Float64(Float64(z / t_3) * Float64(t - a)))
	tmp = 0.0
	if (t_2 <= -2e-184)
		tmp = t_4;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = t_4;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision] - N[(N[((-y) * N[(x / N[(b - y), $MachinePrecision]), $MachinePrecision] + N[(N[(y / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * x), $MachinePrecision] - N[(N[(a - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y / t$95$3), $MachinePrecision] * x + N[(N[(z / t$95$3), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-184], t$95$4, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, Infinity], t$95$4, t$95$1]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -2.0000000000000001e-184 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 77.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. div-sub N/A

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

   5. Applied rewrites 97.7%

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

   if -2.0000000000000001e-184 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 9.2%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\frac{t}{b - y} + \frac{x \cdot y}{z \cdot \left(b - y\right)}\right) - \left(\frac{a}{b - y} + \frac{y \cdot \left(t - a\right)}{z \cdot {\left(b - y\right)}^{2}}\right)} \]

   5. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\frac{t - a}{b - y} - \frac{\mathsf{fma}\left(-y, \frac{x}{b - y}, \left(t - a\right) \cdot \frac{y}{{\left(b - y\right)}^{2}}\right)}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot x - \left(a - t\right) \cdot z}{\left(b - y\right) \cdot z + y} \leq -2 \cdot 10^{-184}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(b - y, z, y\right)}, x, \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \cdot \left(t - a\right)\right)\\ \mathbf{elif}\;\frac{y \cdot x - \left(a - t\right) \cdot z}{\left(b - y\right) \cdot z + y} \leq 0:\\ \;\;\;\;\frac{a - t}{y - b} - \frac{\mathsf{fma}\left(-y, \frac{x}{b - y}, \frac{y}{{\left(b - y\right)}^{2}} \cdot \left(t - a\right)\right)}{z}\\ \mathbf{elif}\;\frac{y \cdot x - \left(a - t\right) \cdot z}{\left(b - y\right) \cdot z + y} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(b - y, z, y\right)}, x, \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \cdot \left(t - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b} - \frac{\mathsf{fma}\left(-y, \frac{x}{b - y}, \frac{y}{{\left(b - y\right)}^{2}} \cdot \left(t - a\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ t_2 := \mathsf{fma}\left(z, b - y, y\right)\\ t_3 := \frac{y \cdot x - \left(a - t\right) \cdot z}{\left(b - y\right) \cdot z + y}\\ t_4 := \mathsf{fma}\left(y, \frac{x}{t\_2}, \frac{z}{t\_2} \cdot \left(t - a\right)\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-273}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 10^{+300}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b)))
        (t_2 (fma z (- b y) y))
        (t_3 (/ (- (* y x) (* (- a t) z)) (+ (* (- b y) z) y)))
        (t_4 (fma y (/ x t_2) (* (/ z t_2) (- t a)))))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -5e-273)
       t_3
       (if (<= t_3 0.0)
         t_1
         (if (<= t_3 1e+300) t_3 (if (<= t_3 INFINITY) t_4 t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / (y - b);
	double t_2 = fma(z, (b - y), y);
	double t_3 = ((y * x) - ((a - t) * z)) / (((b - y) * z) + y);
	double t_4 = fma(y, (x / t_2), ((z / t_2) * (t - a)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -5e-273) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = t_1;
	} else if (t_3 <= 1e+300) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - t) / Float64(y - b))
	t_2 = fma(z, Float64(b - y), y)
	t_3 = Float64(Float64(Float64(y * x) - Float64(Float64(a - t) * z)) / Float64(Float64(Float64(b - y) * z) + y))
	t_4 = fma(y, Float64(x / t_2), Float64(Float64(z / t_2) * Float64(t - a)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -5e-273)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = t_1;
	elseif (t_3 <= 1e+300)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(y * x), $MachinePrecision] - N[(N[(a - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(y * N[(x / t$95$2), $MachinePrecision] + N[(N[(z / t$95$2), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -5e-273], t$95$3, If[LessEqual[t$95$3, 0.0], t$95$1, If[LessEqual[t$95$3, 1e+300], t$95$3, If[LessEqual[t$95$3, Infinity], t$95$4, t$95$1]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 1.0000000000000001e300 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 24.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. div-sub N/A

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 97.9%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -4.99999999999999965e-273 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.0000000000000001e300

   1. Initial program 99.6%

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

   if -4.99999999999999965e-273 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 7.8%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 67.6

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Applied rewrites 67.6%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot x - \left(a - t\right) \cdot z}{\left(b - y\right) \cdot z + y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{\mathsf{fma}\left(z, b - y, y\right)}, \frac{z}{\mathsf{fma}\left(z, b - y, y\right)} \cdot \left(t - a\right)\right)\\ \mathbf{elif}\;\frac{y \cdot x - \left(a - t\right) \cdot z}{\left(b - y\right) \cdot z + y} \leq -5 \cdot 10^{-273}:\\ \;\;\;\;\frac{y \cdot x - \left(a - t\right) \cdot z}{\left(b - y\right) \cdot z + y}\\ \mathbf{elif}\;\frac{y \cdot x - \left(a - t\right) \cdot z}{\left(b - y\right) \cdot z + y} \leq 0:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;\frac{y \cdot x - \left(a - t\right) \cdot z}{\left(b - y\right) \cdot z + y} \leq 10^{+300}:\\ \;\;\;\;\frac{y \cdot x - \left(a - t\right) \cdot z}{\left(b - y\right) \cdot z + y}\\ \mathbf{elif}\;\frac{y \cdot x - \left(a - t\right) \cdot z}{\left(b - y\right) \cdot z + y} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{\mathsf{fma}\left(z, b - y, y\right)}, \frac{z}{\mathsf{fma}\left(z, b - y, y\right)} \cdot \left(t - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -9.6 \cdot 10^{+69}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t\_1}, x, \frac{z}{t\_1} \cdot \left(t - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y)) (t_2 (/ (- a t) (- y b))))
   (if (<= z -9.6e+69)
     t_2
     (if (<= z 7.2e+163) (fma (/ y t_1) x (* (/ z t_1) (- t a))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b - y), z, y);
	double t_2 = (a - t) / (y - b);
	double tmp;
	if (z <= -9.6e+69) {
		tmp = t_2;
	} else if (z <= 7.2e+163) {
		tmp = fma((y / t_1), x, ((z / t_1) * (t - a)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(a - t) / Float64(y - b))
	tmp = 0.0
	if (z <= -9.6e+69)
		tmp = t_2;
	elseif (z <= 7.2e+163)
		tmp = fma(Float64(y / t_1), x, Float64(Float64(z / t_1) * Float64(t - a)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.6e+69], t$95$2, If[LessEqual[z, 7.2e+163], N[(N[(y / t$95$1), $MachinePrecision] * x + N[(N[(z / t$95$1), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.6000000000000007e69 or 7.19999999999999955e163 < z

   1. Initial program 18.2%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 84.3

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Applied rewrites 84.3%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -9.6000000000000007e69 < z < 7.19999999999999955e163

   1. Initial program 78.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. div-sub N/A

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

   5. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(b - y, z, y\right)}, x, \left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(b - y, z, y\right)}, x, \frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \cdot \left(t - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{y \cdot x - \left(a - t\right) \cdot z}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -5.2e+68)
     t_1
     (if (<= z 9.5e+46)
       (/ (- (* y x) (* (- a t) z)) (+ (* (- b y) z) y))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / (y - b);
	double tmp;
	if (z <= -5.2e+68) {
		tmp = t_1;
	} else if (z <= 9.5e+46) {
		tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + y);
	} 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 = (a - t) / (y - b)
    if (z <= (-5.2d+68)) then
        tmp = t_1
    else if (z <= 9.5d+46) then
        tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + y)
    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 = (a - t) / (y - b);
	double tmp;
	if (z <= -5.2e+68) {
		tmp = t_1;
	} else if (z <= 9.5e+46) {
		tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - t) / (y - b)
	tmp = 0
	if z <= -5.2e+68:
		tmp = t_1
	elif z <= 9.5e+46:
		tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - t) / Float64(y - b))
	tmp = 0.0
	if (z <= -5.2e+68)
		tmp = t_1;
	elseif (z <= 9.5e+46)
		tmp = Float64(Float64(Float64(y * x) - Float64(Float64(a - t) * z)) / Float64(Float64(Float64(b - y) * z) + y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - t) / (y - b);
	tmp = 0.0;
	if (z <= -5.2e+68)
		tmp = t_1;
	elseif (z <= 9.5e+46)
		tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.2e+68], t$95$1, If[LessEqual[z, 9.5e+46], N[(N[(N[(y * x), $MachinePrecision] - N[(N[(a - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.1999999999999996e68 or 9.5000000000000008e46 < z

   1. Initial program 27.5%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 80.6

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Applied rewrites 80.6%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -5.1999999999999996e68 < z < 9.5000000000000008e46

   1. Initial program 83.8%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{y \cdot x - \left(a - t\right) \cdot z}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{+17}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1.24 \cdot 10^{-253}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, a, y \cdot x\right)}{t\_1}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y)) (t_2 (/ (- a t) (- y b))))
   (if (<= z -1.15e+17)
     t_2
     (if (<= z -1.24e-253)
       (/ (fma (- z) a (* y x)) t_1)
       (if (<= z 4.2e-20) (/ (fma t z (* y x)) t_1) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b - y), z, y);
	double t_2 = (a - t) / (y - b);
	double tmp;
	if (z <= -1.15e+17) {
		tmp = t_2;
	} else if (z <= -1.24e-253) {
		tmp = fma(-z, a, (y * x)) / t_1;
	} else if (z <= 4.2e-20) {
		tmp = fma(t, z, (y * x)) / t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(a - t) / Float64(y - b))
	tmp = 0.0
	if (z <= -1.15e+17)
		tmp = t_2;
	elseif (z <= -1.24e-253)
		tmp = Float64(fma(Float64(-z), a, Float64(y * x)) / t_1);
	elseif (z <= 4.2e-20)
		tmp = Float64(fma(t, z, Float64(y * x)) / t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.15e+17], t$95$2, If[LessEqual[z, -1.24e-253], N[(N[((-z) * a + N[(y * x), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 4.2e-20], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.15e17 or 4.1999999999999998e-20 < z

   1. Initial program 36.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 79.2

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -1.15e17 < z < -1.23999999999999995e-253

   1. Initial program 83.6%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 71.4

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

   5. Applied rewrites 71.4%

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

   if -1.23999999999999995e-253 < z < 4.1999999999999998e-20

   1. Initial program 83.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 68.4

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

   5. Applied rewrites 68.4%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+17}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq -1.24 \cdot 10^{-253}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, a, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+17}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-262}:\\ \;\;\;\;\frac{y}{t\_1} \cdot x\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y)) (t_2 (/ (- a t) (- y b))))
   (if (<= z -1.1e+17)
     t_2
     (if (<= z 4.2e-262)
       (* (/ y t_1) x)
       (if (<= z 4.2e-20) (/ (fma t z (* y x)) t_1) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b - y), z, y);
	double t_2 = (a - t) / (y - b);
	double tmp;
	if (z <= -1.1e+17) {
		tmp = t_2;
	} else if (z <= 4.2e-262) {
		tmp = (y / t_1) * x;
	} else if (z <= 4.2e-20) {
		tmp = fma(t, z, (y * x)) / t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(a - t) / Float64(y - b))
	tmp = 0.0
	if (z <= -1.1e+17)
		tmp = t_2;
	elseif (z <= 4.2e-262)
		tmp = Float64(Float64(y / t_1) * x);
	elseif (z <= 4.2e-20)
		tmp = Float64(fma(t, z, Float64(y * x)) / t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.1e+17], t$95$2, If[LessEqual[z, 4.2e-262], N[(N[(y / t$95$1), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 4.2e-20], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1e17 or 4.1999999999999998e-20 < z

   1. Initial program 36.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 79.2

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -1.1e17 < z < 4.1999999999999999e-262

   1. Initial program 81.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 67.2

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

   5. Applied rewrites 67.2%

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

   if 4.1999999999999999e-262 < z < 4.1999999999999998e-20

   1. Initial program 86.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 67.0

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

   5. Applied rewrites 67.0%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+17}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-262}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -1.1e+17)
     t_1
     (if (<= z 1.2e-22) (* (/ y (fma (- b y) z y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / (y - b);
	double tmp;
	if (z <= -1.1e+17) {
		tmp = t_1;
	} else if (z <= 1.2e-22) {
		tmp = (y / fma((b - y), z, y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - t) / Float64(y - b))
	tmp = 0.0
	if (z <= -1.1e+17)
		tmp = t_1;
	elseif (z <= 1.2e-22)
		tmp = Float64(Float64(y / fma(Float64(b - y), z, y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.1e+17], t$95$1, If[LessEqual[z, 1.2e-22], N[(N[(y / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1e17 or 1.20000000000000001e-22 < z

   1. Initial program 36.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 79.2

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

   if -1.1e17 < z < 1.20000000000000001e-22

   1. Initial program 83.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 60.1

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

   5. Applied rewrites 60.1%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+17}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{+125}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2600000000:\\ \;\;\;\;\frac{a}{y - b}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+61}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -1.15e+125)
     t_1
     (if (<= y -2600000000.0)
       (/ a (- y b))
       (if (<= y 6.8e+61) (/ (- t a) b) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -1.15e+125) {
		tmp = t_1;
	} else if (y <= -2600000000.0) {
		tmp = a / (y - b);
	} else if (y <= 6.8e+61) {
		tmp = (t - a) / 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 = x / (1.0d0 - z)
    if (y <= (-1.15d+125)) then
        tmp = t_1
    else if (y <= (-2600000000.0d0)) then
        tmp = a / (y - b)
    else if (y <= 6.8d+61) then
        tmp = (t - a) / 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 = x / (1.0 - z);
	double tmp;
	if (y <= -1.15e+125) {
		tmp = t_1;
	} else if (y <= -2600000000.0) {
		tmp = a / (y - b);
	} else if (y <= 6.8e+61) {
		tmp = (t - a) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 - z)
	tmp = 0
	if y <= -1.15e+125:
		tmp = t_1
	elif y <= -2600000000.0:
		tmp = a / (y - b)
	elif y <= 6.8e+61:
		tmp = (t - a) / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 - z))
	tmp = 0.0
	if (y <= -1.15e+125)
		tmp = t_1;
	elseif (y <= -2600000000.0)
		tmp = Float64(a / Float64(y - b));
	elseif (y <= 6.8e+61)
		tmp = Float64(Float64(t - a) / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 - z);
	tmp = 0.0;
	if (y <= -1.15e+125)
		tmp = t_1;
	elseif (y <= -2600000000.0)
		tmp = a / (y - b);
	elseif (y <= 6.8e+61)
		tmp = (t - a) / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.15e+125], t$95$1, If[LessEqual[y, -2600000000.0], N[(a / N[(y - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.8e+61], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.15000000000000006e125 or 6.80000000000000051e61 < y

   1. Initial program 41.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{x}{1 + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}} \]

      3. unsub-neg N/A

         \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

      4. lower--.f64 64.7

         \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

   5. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\frac{x}{1 - z}} \]

   if -1.15000000000000006e125 < y < -2.6e9

   1. Initial program 42.3%

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

   3. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. neg-mul-1 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \frac{z}{y + z \cdot \left(b - y\right)} \]

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 29.7

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

   5. Applied rewrites 29.7%

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

   6. Taylor expanded in z around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{a}{b - y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.0%

      \[\leadsto \frac{-a}{\color{blue}{b - y}} \]

   if -2.6e9 < y < 6.80000000000000051e61

   1. Initial program 75.1%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{t - a}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b}} \]

      2. lower--.f64 58.1

         \[\leadsto \frac{\color{blue}{t - a}}{b} \]

   5. Applied rewrites 58.1%

      \[\leadsto \color{blue}{\frac{t - a}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+125}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{elif}\;y \leq -2600000000:\\ \;\;\;\;\frac{a}{y - b}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+61}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{-77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -5.7 \cdot 10^{-269}:\\ \;\;\;\;\frac{-a}{b}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+18}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -1.45e-77)
     t_1
     (if (<= y -5.7e-269) (/ (- a) b) (if (<= y 5.4e+18) (/ t b) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -1.45e-77) {
		tmp = t_1;
	} else if (y <= -5.7e-269) {
		tmp = -a / b;
	} else if (y <= 5.4e+18) {
		tmp = t / 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 = x / (1.0d0 - z)
    if (y <= (-1.45d-77)) then
        tmp = t_1
    else if (y <= (-5.7d-269)) then
        tmp = -a / b
    else if (y <= 5.4d+18) then
        tmp = t / 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 = x / (1.0 - z);
	double tmp;
	if (y <= -1.45e-77) {
		tmp = t_1;
	} else if (y <= -5.7e-269) {
		tmp = -a / b;
	} else if (y <= 5.4e+18) {
		tmp = t / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 - z)
	tmp = 0
	if y <= -1.45e-77:
		tmp = t_1
	elif y <= -5.7e-269:
		tmp = -a / b
	elif y <= 5.4e+18:
		tmp = t / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 - z))
	tmp = 0.0
	if (y <= -1.45e-77)
		tmp = t_1;
	elseif (y <= -5.7e-269)
		tmp = Float64(Float64(-a) / b);
	elseif (y <= 5.4e+18)
		tmp = Float64(t / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 - z);
	tmp = 0.0;
	if (y <= -1.45e-77)
		tmp = t_1;
	elseif (y <= -5.7e-269)
		tmp = -a / b;
	elseif (y <= 5.4e+18)
		tmp = t / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.45e-77], t$95$1, If[LessEqual[y, -5.7e-269], N[((-a) / b), $MachinePrecision], If[LessEqual[y, 5.4e+18], N[(t / b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.4499999999999999e-77 or 5.4e18 < y

   1. Initial program 45.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{x}{1 + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}} \]

      3. unsub-neg N/A

         \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

      4. lower--.f64 49.6

         \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

   5. Applied rewrites 49.6%

      \[\leadsto \color{blue}{\frac{x}{1 - z}} \]

   if -1.4499999999999999e-77 < y < -5.69999999999999969e-269

   1. Initial program 77.3%

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

   3. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. neg-mul-1 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \frac{z}{y + z \cdot \left(b - y\right)} \]

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 46.8

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

   5. Applied rewrites 46.8%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 46.6%

      \[\leadsto \frac{-a}{\color{blue}{b}} \]

   if -5.69999999999999969e-269 < y < 5.4e18

   1. Initial program 77.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 77.8

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

   4. Applied rewrites 77.8%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 53.7

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

   7. Applied rewrites 53.7%

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

   8. Taylor expanded in y around 0

      \[\leadsto \frac{t}{\color{blue}{b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 40.8%

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

Alternative 10: 62.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -1.16 \cdot 10^{+125}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+73}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -1.16e+125) t_1 (if (<= y 3.1e+73) (/ (- a t) (- y b)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -1.16e+125) {
		tmp = t_1;
	} else if (y <= 3.1e+73) {
		tmp = (a - t) / (y - 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 = x / (1.0d0 - z)
    if (y <= (-1.16d+125)) then
        tmp = t_1
    else if (y <= 3.1d+73) then
        tmp = (a - t) / (y - 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 = x / (1.0 - z);
	double tmp;
	if (y <= -1.16e+125) {
		tmp = t_1;
	} else if (y <= 3.1e+73) {
		tmp = (a - t) / (y - b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 - z)
	tmp = 0
	if y <= -1.16e+125:
		tmp = t_1
	elif y <= 3.1e+73:
		tmp = (a - t) / (y - b)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 - z))
	tmp = 0.0
	if (y <= -1.16e+125)
		tmp = t_1;
	elseif (y <= 3.1e+73)
		tmp = Float64(Float64(a - t) / Float64(y - b));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 - z);
	tmp = 0.0;
	if (y <= -1.16e+125)
		tmp = t_1;
	elseif (y <= 3.1e+73)
		tmp = (a - t) / (y - b);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.16e+125], t$95$1, If[LessEqual[y, 3.1e+73], N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.16000000000000009e125 or 3.1e73 < y

   1. Initial program 41.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{x}{1 + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}} \]

      3. unsub-neg N/A

         \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

      4. lower--.f64 64.7

         \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

   5. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\frac{x}{1 - z}} \]

   if -1.16000000000000009e125 < y < 3.1e73

   1. Initial program 70.6%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{t - a}}{b - y} \]

      3. lower--.f64 66.6

         \[\leadsto \frac{t - a}{\color{blue}{b - y}} \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{+125}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+73}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+61}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -2.2e+91) t_1 (if (<= y 6.8e+61) (/ (- t a) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -2.2e+91) {
		tmp = t_1;
	} else if (y <= 6.8e+61) {
		tmp = (t - a) / 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 = x / (1.0d0 - z)
    if (y <= (-2.2d+91)) then
        tmp = t_1
    else if (y <= 6.8d+61) then
        tmp = (t - a) / 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 = x / (1.0 - z);
	double tmp;
	if (y <= -2.2e+91) {
		tmp = t_1;
	} else if (y <= 6.8e+61) {
		tmp = (t - a) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 - z)
	tmp = 0
	if y <= -2.2e+91:
		tmp = t_1
	elif y <= 6.8e+61:
		tmp = (t - a) / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 - z))
	tmp = 0.0
	if (y <= -2.2e+91)
		tmp = t_1;
	elseif (y <= 6.8e+61)
		tmp = Float64(Float64(t - a) / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 - z);
	tmp = 0.0;
	if (y <= -2.2e+91)
		tmp = t_1;
	elseif (y <= 6.8e+61)
		tmp = (t - a) / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.2e+91], t$95$1, If[LessEqual[y, 6.8e+61], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.19999999999999999e91 or 6.80000000000000051e61 < y

   1. Initial program 42.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{x}{1 + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}} \]

      3. unsub-neg N/A

         \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

      4. lower--.f64 61.3

         \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

   5. Applied rewrites 61.3%

      \[\leadsto \color{blue}{\frac{x}{1 - z}} \]

   if -2.19999999999999999e91 < y < 6.80000000000000051e61

   1. Initial program 71.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{t - a}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - a}{b}} \]

      2. lower--.f64 56.4

         \[\leadsto \frac{\color{blue}{t - a}}{b} \]

   5. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\frac{t - a}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 37.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{-a}{b}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2.1e-9) (/ (- a) b) (if (<= z 4.6e-20) (fma z x x) (/ t b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2.1e-9) {
		tmp = -a / b;
	} else if (z <= 4.6e-20) {
		tmp = fma(z, x, x);
	} else {
		tmp = t / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -2.1e-9)
		tmp = Float64(Float64(-a) / b);
	elseif (z <= 4.6e-20)
		tmp = fma(z, x, x);
	else
		tmp = Float64(t / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2.1e-9], N[((-a) / b), $MachinePrecision], If[LessEqual[z, 4.6e-20], N[(z * x + x), $MachinePrecision], N[(t / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.10000000000000019e-9

   1. Initial program 31.8%

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

   3. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. neg-mul-1 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \frac{z}{y + z \cdot \left(b - y\right)} \]

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 31.2

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

   5. Applied rewrites 31.2%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 24.1%

      \[\leadsto \frac{-a}{\color{blue}{b}} \]

   if -2.10000000000000019e-9 < z < 4.5999999999999998e-20

   1. Initial program 83.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{x}{1 + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}} \]

      3. unsub-neg N/A

         \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

      4. lower--.f64 48.6

         \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

   5. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\frac{x}{1 - z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{x \cdot z} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.6%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{x}, x\right) \]

   if 4.5999999999999998e-20 < z

   1. Initial program 43.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 43.3

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 26.7

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

   7. Applied rewrites 26.7%

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

   8. Taylor expanded in y around 0

      \[\leadsto \frac{t}{\color{blue}{b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 33.5%

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

Alternative 13: 37.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.25:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2.25) (/ t b) (if (<= z 4.6e-20) (fma z x x) (/ t b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2.25) {
		tmp = t / b;
	} else if (z <= 4.6e-20) {
		tmp = fma(z, x, x);
	} else {
		tmp = t / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -2.25)
		tmp = Float64(t / b);
	elseif (z <= 4.6e-20)
		tmp = fma(z, x, x);
	else
		tmp = Float64(t / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2.25], N[(t / b), $MachinePrecision], If[LessEqual[z, 4.6e-20], N[(z * x + x), $MachinePrecision], N[(t / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.25 or 4.5999999999999998e-20 < z

   1. Initial program 37.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 37.4

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

   4. Applied rewrites 37.4%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 24.0

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

   7. Applied rewrites 24.0%

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

   8. Taylor expanded in y around 0

      \[\leadsto \frac{t}{\color{blue}{b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 29.2%

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

   if -2.25 < z < 4.5999999999999998e-20

   1. Initial program 83.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{x}{1 + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}} \]

      3. unsub-neg N/A

         \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

      4. lower--.f64 47.9

         \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

   5. Applied rewrites 47.9%

      \[\leadsto \color{blue}{\frac{x}{1 - z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{x \cdot z} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.9%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 25.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, x, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(z * x + x), $MachinePrecision]

Derivation

1. Initial program 59.8%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. mul-1-neg N/A

      \[\leadsto \frac{x}{1 + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}} \]

   3. unsub-neg N/A

      \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

   4. lower--.f64 33.0

      \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

5. Applied rewrites 33.0%

   \[\leadsto \color{blue}{\frac{x}{1 - z}} \]

6. Taylor expanded in z around 0

   \[\leadsto x + \color{blue}{x \cdot z} \]
7. Step-by-step derivation

8. Applied rewrites 25.1%

   \[\leadsto \mathsf{fma}\left(z, \color{blue}{x}, x\right) \]
9. Add Preprocessing

Alternative 15: 25.1% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

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

C

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

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 = 1.0d0 * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 59.8%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. clear-num N/A

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

   8. flip-- N/A

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

   9. lift--.f64 N/A

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

   10. lower-/.f64 59.8

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

4. Applied rewrites 59.8%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

7. Applied rewrites 64.0%

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

8. Taylor expanded in z around 0

   \[\leadsto 1 \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 24.9%

    \[\leadsto 1 \cdot x \]
11. Add Preprocessing

Alternative 16: 3.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ z \cdot x \end{array} \]

FPCore

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

C

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

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 = z * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.8%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. mul-1-neg N/A

      \[\leadsto \frac{x}{1 + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}} \]

   3. unsub-neg N/A

      \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

   4. lower--.f64 33.0

      \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

5. Applied rewrites 33.0%

   \[\leadsto \color{blue}{\frac{x}{1 - z}} \]

6. Taylor expanded in z around 0

   \[\leadsto x + \color{blue}{x \cdot z} \]
7. Step-by-step derivation

8. Applied rewrites 25.1%

   \[\leadsto \mathsf{fma}\left(z, \color{blue}{x}, x\right) \]

9. Taylor expanded in z around inf

   \[\leadsto x \cdot z \]
10. Step-by-step derivation

11. Applied rewrites 3.5%

    \[\leadsto z \cdot x \]
12. Add Preprocessing

Developer Target 1: 74.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

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 = (((z * t) + (y * x)) / (y + (z * (b - y)))) - (a / ((b - y) + (y / z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024255 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

  :alt
  (! :herbie-platform default (- (/ (+ (* z t) (* y x)) (+ y (* z (- b y)))) (/ a (+ (- b y) (/ y z)))))

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