Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.0% → 94.7%

Time: 15.6s

Alternatives: 14

Speedup: 1.3×

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: 66.0% 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: 94.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y} - \frac{\mathsf{fma}\left(y, \frac{t - a}{\left(b - y\right) \cdot \left(b - y\right)}, x \cdot \frac{y}{y - b}\right)}{z}\\ t_2 := \mathsf{fma}\left(z, b - y, y\right)\\ \mathbf{if}\;z \leq -1.42 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{t\_2}, \frac{z \cdot \left(t - a\right)}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (-
          (/ (- t a) (- b y))
          (/ (fma y (/ (- t a) (* (- b y) (- b y))) (* x (/ y (- y b)))) z)))
        (t_2 (fma z (- b y) y)))
   (if (<= z -1.42e+51)
     t_1
     (if (<= z 1.22e+15) (fma x (/ y t_2) (/ (* z (- t a)) t_2)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((t - a) / (b - y)) - (fma(y, ((t - a) / ((b - y) * (b - y))), (x * (y / (y - b)))) / z);
	double t_2 = fma(z, (b - y), y);
	double tmp;
	if (z <= -1.42e+51) {
		tmp = t_1;
	} else if (z <= 1.22e+15) {
		tmp = fma(x, (y / t_2), ((z * (t - a)) / t_2));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(t - a) / Float64(b - y)) - Float64(fma(y, Float64(Float64(t - a) / Float64(Float64(b - y) * Float64(b - y))), Float64(x * Float64(y / Float64(y - b)))) / z))
	t_2 = fma(z, Float64(b - y), y)
	tmp = 0.0
	if (z <= -1.42e+51)
		tmp = t_1;
	elseif (z <= 1.22e+15)
		tmp = fma(x, Float64(y / t_2), Float64(Float64(z * Float64(t - a)) / t_2));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(N[(y * N[(N[(t - a), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y / N[(y - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[z, -1.42e+51], t$95$1, If[LessEqual[z, 1.22e+15], N[(x * N[(y / t$95$2), $MachinePrecision] + N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.41999999999999998e51 or 1.22e15 < z

   1. Initial program 40.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. --lowering--.f64 44.0

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

   5. Simplified 44.0%

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

   6. Taylor expanded in z around -inf

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

      1. associate--l+ N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{x \cdot y}{b - y} + \frac{y \cdot \left(t - a\right)}{{\left(b - y\right)}^{2}}}{z} + \left(\frac{t}{b - y} - \frac{a}{b - y}\right)} \]

      2. div-sub N/A

         \[\leadsto -1 \cdot \frac{-1 \cdot \frac{x \cdot y}{b - y} + \frac{y \cdot \left(t - a\right)}{{\left(b - y\right)}^{2}}}{z} + \color{blue}{\frac{t - a}{b - y}} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{x \cdot y}{b - y} + \frac{y \cdot \left(t - a\right)}{{\left(b - y\right)}^{2}}}{z} + \frac{t - a}{b - y}} \]

   8. Simplified 93.2%

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

   if -1.41999999999999998e51 < z < 1.22e15

   1. Initial program 85.4%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. --lowering--.f64 98.2

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

   5. Simplified 98.2%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.42 \cdot 10^{+51}:\\ \;\;\;\;\frac{t - a}{b - y} - \frac{\mathsf{fma}\left(y, \frac{t - a}{\left(b - y\right) \cdot \left(b - y\right)}, x \cdot \frac{y}{y - b}\right)}{z}\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, \frac{z \cdot \left(t - a\right)}{\mathsf{fma}\left(z, b - y, y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y} - \frac{\mathsf{fma}\left(y, \frac{t - a}{\left(b - y\right) \cdot \left(b - y\right)}, x \cdot \frac{y}{y - b}\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot \left(b - y\right)}\\ t_3 := \mathsf{fma}\left(z, b - y, y\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+273}:\\ \;\;\;\;\mathsf{fma}\left(t - a, \frac{z}{t\_3}, x\right)\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-304}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{t\_3}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (/ (+ (* z (- t a)) (* y x)) (+ y (* z (- b y)))))
        (t_3 (fma z (- b y) y)))
   (if (<= t_2 -2e+273)
     (fma (- t a) (/ z t_3) x)
     (if (<= t_2 -5e-304)
       t_2
       (if (<= t_2 0.0)
         t_1
         (if (<= t_2 2e+299)
           t_2
           (if (<= t_2 INFINITY) (fma (/ (- t a) t_3) z x) t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double t_2 = ((z * (t - a)) + (y * x)) / (y + (z * (b - y)));
	double t_3 = fma(z, (b - y), y);
	double tmp;
	if (t_2 <= -2e+273) {
		tmp = fma((t - a), (z / t_3), x);
	} else if (t_2 <= -5e-304) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 2e+299) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fma(((t - a) / t_3), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = Float64(Float64(Float64(z * Float64(t - a)) + Float64(y * x)) / Float64(y + Float64(z * Float64(b - y))))
	t_3 = fma(z, Float64(b - y), y)
	tmp = 0.0
	if (t_2 <= -2e+273)
		tmp = fma(Float64(t - a), Float64(z / t_3), x);
	elseif (t_2 <= -5e-304)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 2e+299)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = fma(Float64(Float64(t - a) / t_3), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+273], N[(N[(t - a), $MachinePrecision] * N[(z / t$95$3), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$2, -5e-304], t$95$2, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 2e+299], t$95$2, If[LessEqual[t$95$2, Infinity], N[(N[(N[(t - a), $MachinePrecision] / t$95$3), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.99999999999999989e273

   1. Initial program 42.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. --lowering--.f64 84.6

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

   5. Simplified 84.6%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. associate-*r/ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. --lowering--.f64 57.3

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

   7. Applied egg-rr 57.3%

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

   8. Taylor expanded in z around 0

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

   10. Simplified 87.2%

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

   if -1.99999999999999989e273 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -4.99999999999999965e-304 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 2.0000000000000001e299

   1. Initial program 99.3%

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

   if -4.99999999999999965e-304 < (/.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 8.9%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. --lowering--.f64 77.1

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

   5. Simplified 77.1%

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

   if 2.0000000000000001e299 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 29.2%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. --lowering--.f64 67.4

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

   5. Simplified 67.4%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. associate-*r/ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. --lowering--.f64 61.5

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

   7. Applied egg-rr 61.5%

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

   8. Taylor expanded in z around 0

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

   10. Simplified 77.2%

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

       1. clear-num N/A

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

       2. associate-*r/ N/A

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

       3. div-inv N/A

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

       4. times-frac N/A

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

       5. clear-num N/A

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

       6. /-rgt-identity N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. /-lowering-/.f64 N/A

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

       9. --lowering--.f64 N/A

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

       10. accelerator-lowering-fma.f64 N/A

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

       11. --lowering--.f64 77.3

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

   12. Applied egg-rr 77.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, z, x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot \left(b - y\right)} \leq -2 \cdot 10^{+273}:\\ \;\;\;\;\mathsf{fma}\left(t - a, \frac{z}{\mathsf{fma}\left(z, b - y, y\right)}, x\right)\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot \left(b - y\right)} \leq -5 \cdot 10^{-304}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot \left(b - y\right)} \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -2.75 \cdot 10^{+51}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{t\_1}, \frac{z \cdot \left(t - a\right)}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- b y) y)) (t_2 (/ (- t a) (- b y))))
   (if (<= z -2.75e+51)
     t_2
     (if (<= z 1.25e+50) (fma x (/ y t_1) (/ (* z (- t a)) t_1)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (b - y), y);
	double t_2 = (t - a) / (b - y);
	double tmp;
	if (z <= -2.75e+51) {
		tmp = t_2;
	} else if (z <= 1.25e+50) {
		tmp = fma(x, (y / t_1), ((z * (t - a)) / t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(b - y), y)
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -2.75e+51)
		tmp = t_2;
	elseif (z <= 1.25e+50)
		tmp = fma(x, Float64(y / t_1), Float64(Float64(z * Float64(t - a)) / t_1));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.75e+51], t$95$2, If[LessEqual[z, 1.25e+50], N[(x * N[(y / t$95$1), $MachinePrecision] + N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.75e51 or 1.25e50 < z

   1. Initial program 38.1%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. --lowering--.f64 85.6

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

   5. Simplified 85.6%

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

   if -2.75e51 < z < 1.25e50

   1. Initial program 84.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. --lowering--.f64 97.7

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

   5. Simplified 97.7%

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

Alternative 4: 80.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(t - a, \frac{z}{\mathsf{fma}\left(z, b - y, y\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -4e+23)
     t_1
     (if (<= z 1.1e+15) (fma (- t a) (/ z (fma z (- b y) y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -4e+23) {
		tmp = t_1;
	} else if (z <= 1.1e+15) {
		tmp = fma((t - a), (z / fma(z, (b - y), y)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -4e+23)
		tmp = t_1;
	elseif (z <= 1.1e+15)
		tmp = fma(Float64(t - a), Float64(z / fma(z, Float64(b - y), y)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4e+23], t$95$1, If[LessEqual[z, 1.1e+15], N[(N[(t - a), $MachinePrecision] * N[(z / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.9999999999999997e23 or 1.1e15 < z

   1. Initial program 42.0%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. --lowering--.f64 83.7

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

   5. Simplified 83.7%

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

   if -3.9999999999999997e23 < z < 1.1e15

   1. Initial program 85.5%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. --lowering--.f64 98.2

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

   5. Simplified 98.2%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. associate-*r/ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. --lowering--.f64 82.9

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

   7. Applied egg-rr 82.9%

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

   8. Taylor expanded in z around 0

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

   10. Simplified 83.1%

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

Alternative 5: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -0.00195:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t - a, \frac{z}{\mathsf{fma}\left(z, b, y\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -0.00195)
     t_1
     (if (<= z 1.12e-9) (fma (- t a) (/ z (fma z b y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -0.00195) {
		tmp = t_1;
	} else if (z <= 1.12e-9) {
		tmp = fma((t - a), (z / fma(z, b, y)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -0.00195)
		tmp = t_1;
	elseif (z <= 1.12e-9)
		tmp = fma(Float64(t - a), Float64(z / fma(z, b, y)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.00195], t$95$1, If[LessEqual[z, 1.12e-9], N[(N[(t - a), $MachinePrecision] * N[(z / N[(z * b + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.0019499999999999999 or 1.12000000000000006e-9 < z

   1. Initial program 44.3%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. --lowering--.f64 83.1

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

   5. Simplified 83.1%

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

   if -0.0019499999999999999 < z < 1.12000000000000006e-9

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. --lowering--.f64 98.1

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

   5. Simplified 98.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. associate-*r/ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. --lowering--.f64 82.8

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

   7. Applied egg-rr 82.8%

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

   8. Taylor expanded in z around 0

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

   10. Simplified 83.0%

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

   11. Taylor expanded in b around inf

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

   13. Simplified 82.5%

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

Alternative 6: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -6 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-201}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t}{y}, x\right)\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{a}{-y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -6e-26)
     t_1
     (if (<= z 3.1e-201)
       (fma z (/ t y) x)
       (if (<= z 2.25e-10) (fma z (/ a (- y)) x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -6e-26) {
		tmp = t_1;
	} else if (z <= 3.1e-201) {
		tmp = fma(z, (t / y), x);
	} else if (z <= 2.25e-10) {
		tmp = fma(z, (a / -y), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -6e-26)
		tmp = t_1;
	elseif (z <= 3.1e-201)
		tmp = fma(z, Float64(t / y), x);
	elseif (z <= 2.25e-10)
		tmp = fma(z, Float64(a / Float64(-y)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6e-26], t$95$1, If[LessEqual[z, 3.1e-201], N[(z * N[(t / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 2.25e-10], N[(z * N[(a / (-y)), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.00000000000000023e-26 or 2.25e-10 < z

   1. Initial program 47.0%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. --lowering--.f64 80.9

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

   5. Simplified 80.9%

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

   if -6.00000000000000023e-26 < z < 3.0999999999999999e-201

   1. Initial program 86.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. --lowering--.f64 98.3

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

   5. Simplified 98.3%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate--r+ N/A

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

      4. --lowering--.f64 N/A

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

      5. div-sub N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 54.6

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

   8. Simplified 54.6%

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

   9. Taylor expanded in t around inf

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

       1. /-lowering-/.f64 66.2

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

   11. Simplified 66.2%

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

   if 3.0999999999999999e-201 < z < 2.25e-10

   1. Initial program 80.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. --lowering--.f64 97.4

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

   5. Simplified 97.4%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate--r+ N/A

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

      4. --lowering--.f64 N/A

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

      5. div-sub N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 44.2

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

   8. Simplified 44.2%

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

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{-1 \cdot \frac{a}{y}}, x\right) \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

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

       2. mul-1-neg N/A

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

       3. /-lowering-/.f64 N/A

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

       4. neg-lowering-neg.f64 59.7

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

   11. Simplified 59.7%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-26}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-201}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t}{y}, x\right)\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{a}{-y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.14 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(t - a, \frac{z}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.14e-8) t_1 (if (<= z 6.6e-10) (fma (- t a) (/ z y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.14e-8) {
		tmp = t_1;
	} else if (z <= 6.6e-10) {
		tmp = fma((t - a), (z / y), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1.14e-8)
		tmp = t_1;
	elseif (z <= 6.6e-10)
		tmp = fma(Float64(t - a), Float64(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[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.14e-8], t$95$1, If[LessEqual[z, 6.6e-10], N[(N[(t - a), $MachinePrecision] * N[(z / y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.14e-8 or 6.6e-10 < z

   1. Initial program 44.3%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. --lowering--.f64 83.1

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

   5. Simplified 83.1%

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

   if -1.14e-8 < z < 6.6e-10

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. --lowering--.f64 98.1

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

   5. Simplified 98.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. associate-*r/ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. --lowering--.f64 82.8

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

   7. Applied egg-rr 82.8%

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

   8. Taylor expanded in z around 0

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

   10. Simplified 83.0%

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

   11. Taylor expanded in z around 0

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

       1. /-lowering-/.f64 73.0

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

   13. Simplified 73.0%

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

Alternative 8: 67.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.22 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.22e-26) t_1 (if (<= z 3.4e-33) (fma z (/ t y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.22e-26) {
		tmp = t_1;
	} else if (z <= 3.4e-33) {
		tmp = fma(z, (t / y), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1.22e-26)
		tmp = t_1;
	elseif (z <= 3.4e-33)
		tmp = fma(z, Float64(t / y), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.22e-26], t$95$1, If[LessEqual[z, 3.4e-33], N[(z * N[(t / y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.22e-26 or 3.4000000000000001e-33 < z

   1. Initial program 47.6%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. --lowering--.f64 78.5

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

   5. Simplified 78.5%

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

   if -1.22e-26 < z < 3.4000000000000001e-33

   1. Initial program 85.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. --lowering--.f64 98.0

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

   5. Simplified 98.0%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate--r+ N/A

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

      4. --lowering--.f64 N/A

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

      5. div-sub N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 51.4

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

   8. Simplified 51.4%

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

   9. Taylor expanded in t around inf

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

       1. /-lowering-/.f64 61.2

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

   11. Simplified 61.2%

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

Alternative 9: 64.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{-27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.45e-27) t_1 (if (<= z 3.7e-10) (fma z x x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.45e-27) {
		tmp = t_1;
	} else if (z <= 3.7e-10) {
		tmp = fma(z, x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1.45e-27)
		tmp = t_1;
	elseif (z <= 3.7e-10)
		tmp = fma(z, x, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.45e-27], t$95$1, If[LessEqual[z, 3.7e-10], N[(z * x + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.45000000000000002e-27 or 3.70000000000000015e-10 < z

   1. Initial program 47.0%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. --lowering--.f64 80.9

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

   5. Simplified 80.9%

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

   if -1.45000000000000002e-27 < z < 3.70000000000000015e-10

   1. Initial program 85.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. --lowering--.f64 98.1

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

   5. Simplified 98.1%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate--r+ N/A

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

      4. --lowering--.f64 N/A

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

      5. div-sub N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 51.6

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

   8. Simplified 51.6%

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

   9. Taylor expanded in y around inf

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 51.3

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

   11. Simplified 51.3%

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

Alternative 10: 54.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{-69}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+53}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -1.7e-69) t_1 (if (<= y 3.3e+53) (/ (- 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 <= -1.7e-69) {
		tmp = t_1;
	} else if (y <= 3.3e+53) {
		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 <= (-1.7d-69)) then
        tmp = t_1
    else if (y <= 3.3d+53) 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 <= -1.7e-69) {
		tmp = t_1;
	} else if (y <= 3.3e+53) {
		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 <= -1.7e-69:
		tmp = t_1
	elif y <= 3.3e+53:
		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 <= -1.7e-69)
		tmp = t_1;
	elseif (y <= 3.3e+53)
		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 <= -1.7e-69)
		tmp = t_1;
	elseif (y <= 3.3e+53)
		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, -1.7e-69], t$95$1, If[LessEqual[y, 3.3e+53], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.70000000000000004e-69 or 3.3000000000000002e53 < y

   1. Initial program 53.9%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. /-lowering-/.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. --lowering--.f64 55.8

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

   5. Simplified 55.8%

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

   if -1.70000000000000004e-69 < y < 3.3000000000000002e53

   1. Initial program 80.2%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 54.4

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

   5. Simplified 54.4%

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

Alternative 11: 45.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{b - y}\\ \mathbf{if}\;z \leq -1.75 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- b y))))
   (if (<= z -1.75e-29) t_1 (if (<= z 8.4e-10) (fma z x x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -1.75e-29) {
		tmp = t_1;
	} else if (z <= 8.4e-10) {
		tmp = fma(z, x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(b - y))
	tmp = 0.0
	if (z <= -1.75e-29)
		tmp = t_1;
	elseif (z <= 8.4e-10)
		tmp = fma(z, x, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.75e-29], t$95$1, If[LessEqual[z, 8.4e-10], N[(z * x + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7499999999999999e-29 or 8.3999999999999999e-10 < z

   1. Initial program 47.0%

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

   3. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --lowering--.f64 24.0

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

   5. Simplified 24.0%

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

   6. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 39.6

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

   8. Simplified 39.6%

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

   if -1.7499999999999999e-29 < z < 8.3999999999999999e-10

   1. Initial program 85.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. --lowering--.f64 98.1

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

   5. Simplified 98.1%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate--r+ N/A

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

      4. --lowering--.f64 N/A

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

      5. div-sub N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 51.6

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

   8. Simplified 51.6%

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

   9. Taylor expanded in y around inf

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 51.3

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

   11. Simplified 51.3%

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

Alternative 12: 37.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-15}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.1e-15) (/ t b) (if (<= z 2.15e-10) (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 <= -1.1e-15) {
		tmp = t / b;
	} else if (z <= 2.15e-10) {
		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 <= -1.1e-15)
		tmp = Float64(t / b);
	elseif (z <= 2.15e-10)
		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, -1.1e-15], N[(t / b), $MachinePrecision], If[LessEqual[z, 2.15e-10], N[(z * x + x), $MachinePrecision], N[(t / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.09999999999999993e-15 or 2.15000000000000007e-10 < z

   1. Initial program 45.7%

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

   3. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --lowering--.f64 24.5

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

   5. Simplified 24.5%

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

   6. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 27.9

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

   8. Simplified 27.9%

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

   if -1.09999999999999993e-15 < z < 2.15000000000000007e-10

   1. Initial program 85.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. --lowering--.f64 98.1

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

   5. Simplified 98.1%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate--r+ N/A

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

      4. --lowering--.f64 N/A

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

      5. div-sub N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 51.9

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

   8. Simplified 51.9%

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

   9. Taylor expanded in y around inf

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 50.3

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

   11. Simplified 50.3%

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

Alternative 13: 25.6% 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 67.4%

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

3. Taylor expanded in x around 0

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

   1. associate-/l* N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. +-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. --lowering--.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. --lowering--.f64 N/A

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

   10. +-commutative N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. --lowering--.f64 76.6

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

5. Simplified 76.6%

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

6. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. associate--r+ N/A

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

   4. --lowering--.f64 N/A

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

   5. div-sub N/A

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

   6. /-lowering-/.f64 N/A

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

   7. --lowering--.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 N/A

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

   11. --lowering--.f64 30.1

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

8. Simplified 30.1%

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

9. Taylor expanded in y around inf

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. accelerator-lowering-fma.f64 29.1

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

11. Simplified 29.1%

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

Alternative 14: 25.4% accurate, 39.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return 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 = x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 67.4%

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

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 28.9%

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

Developer Target 1: 73.5% 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 2024199 
(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)))))