Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A

Percentage Accurate: 85.6% → 98.2%

Time: 8.7s

Alternatives: 12

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = x + ((y * (z - t)) / (z - a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = x + ((y * (z - t)) / (z - a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{z - a} + x \leq -5 \cdot 10^{+108}:\\ \;\;\;\;\frac{\frac{y}{z - a}}{\frac{1}{z - t}} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z - a}{z - t}} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (+ (/ (* (- z t) y) (- z a)) x) -5e+108)
   (+ (/ (/ y (- z a)) (/ 1.0 (- z t))) x)
   (+ (/ y (/ (- z a) (- z t))) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((((z - t) * y) / (z - a)) + x) <= -5e+108) {
		tmp = ((y / (z - a)) / (1.0 / (z - t))) + x;
	} else {
		tmp = (y / ((z - a) / (z - t))) + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (((((z - t) * y) / (z - a)) + x) <= (-5d+108)) then
        tmp = ((y / (z - a)) / (1.0d0 / (z - t))) + x
    else
        tmp = (y / ((z - a) / (z - t))) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((((z - t) * y) / (z - a)) + x) <= -5e+108) {
		tmp = ((y / (z - a)) / (1.0 / (z - t))) + x;
	} else {
		tmp = (y / ((z - a) / (z - t))) + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((((z - t) * y) / (z - a)) + x) <= -5e+108:
		tmp = ((y / (z - a)) / (1.0 / (z - t))) + x
	else:
		tmp = (y / ((z - a) / (z - t))) + x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(z - t) * y) / Float64(z - a)) + x) <= -5e+108)
		tmp = Float64(Float64(Float64(y / Float64(z - a)) / Float64(1.0 / Float64(z - t))) + x);
	else
		tmp = Float64(Float64(y / Float64(Float64(z - a) / Float64(z - t))) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((((z - t) * y) / (z - a)) + x) <= -5e+108)
		tmp = ((y / (z - a)) / (1.0 / (z - t))) + x;
	else
		tmp = (y / ((z - a) / (z - t))) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], -5e+108], N[(N[(N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[(N[(z - a), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))) < -4.99999999999999991e108

   1. Initial program 82.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. clear-num N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 99.9

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

   6. Applied rewrites 99.9%

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

   if -4.99999999999999991e108 < (+.f64 x (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)))

   1. Initial program 90.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{z - a} + x \leq -5 \cdot 10^{+108}:\\ \;\;\;\;\frac{\frac{y}{z - a}}{\frac{1}{z - t}} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z - a}{z - t}} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z - a} \cdot \left(z - t\right)\\ t_2 := \frac{\left(z - t\right) \cdot y}{z - a}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-173}:\\ \;\;\;\;\frac{z \cdot y}{z - a} + x\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y (- z a)) (- z t))) (t_2 (/ (* (- z t) y) (- z a))))
   (if (<= t_2 -4e+76)
     t_1
     (if (<= t_2 1e-173)
       (+ (/ (* z y) (- z a)) x)
       (if (<= t_2 2e+111) (fma (/ (- z t) z) y x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / (z - a)) * (z - t);
	double t_2 = ((z - t) * y) / (z - a);
	double tmp;
	if (t_2 <= -4e+76) {
		tmp = t_1;
	} else if (t_2 <= 1e-173) {
		tmp = ((z * y) / (z - a)) + x;
	} else if (t_2 <= 2e+111) {
		tmp = fma(((z - t) / z), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / Float64(z - a)) * Float64(z - t))
	t_2 = Float64(Float64(Float64(z - t) * y) / Float64(z - a))
	tmp = 0.0
	if (t_2 <= -4e+76)
		tmp = t_1;
	elseif (t_2 <= 1e-173)
		tmp = Float64(Float64(Float64(z * y) / Float64(z - a)) + x);
	elseif (t_2 <= 2e+111)
		tmp = fma(Float64(Float64(z - t) / z), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+76], t$95$1, If[LessEqual[t$95$2, 1e-173], N[(N[(N[(z * y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$2, 2e+111], N[(N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < -4.0000000000000002e76 or 1.99999999999999991e111 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))

   1. Initial program 65.8%

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

   3. Taylor expanded in y around inf

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

      1. distribute-lft-out-- N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. distribute-rgt-out-- N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if -4.0000000000000002e76 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < 1e-173

   1. Initial program 99.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 93.5

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

   5. Applied rewrites 93.5%

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

   if 1e-173 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < 1.99999999999999991e111

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 89.8

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

   5. Applied rewrites 89.8%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{z - a} \leq -4 \cdot 10^{+76}:\\ \;\;\;\;\frac{y}{z - a} \cdot \left(z - t\right)\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{z - a} \leq 10^{-173}:\\ \;\;\;\;\frac{z \cdot y}{z - a} + x\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{z - a} \leq 2 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z - a} \cdot \left(z - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z - a} \cdot \left(z - t\right)\\ t_2 := \frac{\left(z - t\right) \cdot y}{z - a}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y (- z a)) (- z t))) (t_2 (/ (* (- z t) y) (- z a))))
   (if (<= t_2 -4e+76)
     t_1
     (if (<= t_2 1e-173)
       (fma (/ z (- z a)) y x)
       (if (<= t_2 2e+111) (fma (/ (- z t) z) y x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / (z - a)) * (z - t);
	double t_2 = ((z - t) * y) / (z - a);
	double tmp;
	if (t_2 <= -4e+76) {
		tmp = t_1;
	} else if (t_2 <= 1e-173) {
		tmp = fma((z / (z - a)), y, x);
	} else if (t_2 <= 2e+111) {
		tmp = fma(((z - t) / z), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / Float64(z - a)) * Float64(z - t))
	t_2 = Float64(Float64(Float64(z - t) * y) / Float64(z - a))
	tmp = 0.0
	if (t_2 <= -4e+76)
		tmp = t_1;
	elseif (t_2 <= 1e-173)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	elseif (t_2 <= 2e+111)
		tmp = fma(Float64(Float64(z - t) / z), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+76], t$95$1, If[LessEqual[t$95$2, 1e-173], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$2, 2e+111], N[(N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < -4.0000000000000002e76 or 1.99999999999999991e111 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))

   1. Initial program 65.8%

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

   3. Taylor expanded in y around inf

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

      1. distribute-lft-out-- N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. distribute-rgt-out-- N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if -4.0000000000000002e76 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < 1e-173

   1. Initial program 99.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 93.5

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

   5. Applied rewrites 93.5%

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

   if 1e-173 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < 1.99999999999999991e111

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 89.8

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

   5. Applied rewrites 89.8%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{z - a} \leq -4 \cdot 10^{+76}:\\ \;\;\;\;\frac{y}{z - a} \cdot \left(z - t\right)\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{z - a} \leq 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{z - a} \leq 2 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z - a} \cdot \left(z - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z - a} \cdot \left(z - t\right)\\ t_2 := \frac{\left(z - t\right) \cdot y}{z - a}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;t\_2 + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y (- z a)) (- z t))) (t_2 (/ (* (- z t) y) (- z a))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 2e+303) (+ t_2 x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / (z - a)) * (z - t);
	double t_2 = ((z - t) * y) / (z - a);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 2e+303) {
		tmp = t_2 + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / (z - a)) * (z - t);
	double t_2 = ((z - t) * y) / (z - a);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 2e+303) {
		tmp = t_2 + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y / (z - a)) * (z - t)
	t_2 = ((z - t) * y) / (z - a)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 2e+303:
		tmp = t_2 + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / Float64(z - a)) * Float64(z - t))
	t_2 = Float64(Float64(Float64(z - t) * y) / Float64(z - a))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 2e+303)
		tmp = Float64(t_2 + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y / (z - a)) * (z - t);
	t_2 = ((z - t) * y) / (z - a);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 2e+303)
		tmp = t_2 + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 2e+303], N[(t$95$2 + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < -inf.0 or 2e303 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))

   1. Initial program 45.9%

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

   3. Taylor expanded in y around inf

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

      1. distribute-lft-out-- N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. distribute-rgt-out-- N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 89.6

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

   5. Applied rewrites 89.6%

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

   if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < 2e303

   1. Initial program 99.4%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{z - a} \leq -\infty:\\ \;\;\;\;\frac{y}{z - a} \cdot \left(z - t\right)\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{z - a} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z - a} \cdot \left(z - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \mathbf{if}\;z \leq -1.12 \cdot 10^{+54}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 130:\\ \;\;\;\;\frac{\left(-t\right) \cdot y}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z t) z) y x)))
   (if (<= z -1.12e+54)
     t_1
     (if (<= z 130.0) (+ (/ (* (- t) y) (- z a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - t) / z), y, x);
	double tmp;
	if (z <= -1.12e+54) {
		tmp = t_1;
	} else if (z <= 130.0) {
		tmp = ((-t * y) / (z - a)) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - t) / z), y, x)
	tmp = 0.0
	if (z <= -1.12e+54)
		tmp = t_1;
	elseif (z <= 130.0)
		tmp = Float64(Float64(Float64(Float64(-t) * y) / Float64(z - a)) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[z, -1.12e+54], t$95$1, If[LessEqual[z, 130.0], N[(N[(N[((-t) * y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.12e54 or 130 < z

   1. Initial program 78.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if -1.12e54 < z < 130

   1. Initial program 94.2%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 85.6

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

   5. Applied rewrites 85.6%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.12 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \mathbf{elif}\;z \leq 130:\\ \;\;\;\;\frac{\left(-t\right) \cdot y}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \mathbf{if}\;z \leq -2 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z t) z) y x)))
   (if (<= z -2e+53) t_1 (if (<= z 3.9e-49) (fma (/ y a) t x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - t) / z), y, x);
	double tmp;
	if (z <= -2e+53) {
		tmp = t_1;
	} else if (z <= 3.9e-49) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - t) / z), y, x)
	tmp = 0.0
	if (z <= -2e+53)
		tmp = t_1;
	elseif (z <= 3.9e-49)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[z, -2e+53], t$95$1, If[LessEqual[z, 3.9e-49], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e53 or 3.90000000000000011e-49 < z

   1. Initial program 81.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 90.9

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

   5. Applied rewrites 90.9%

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

   if -2e53 < z < 3.90000000000000011e-49

   1. Initial program 93.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 80.3

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

   5. Applied rewrites 80.3%

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

Alternative 7: 81.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{if}\;z \leq -1850000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z (- z a)) y x)))
   (if (<= z -1850000000.0) t_1 (if (<= z 2.15e-49) (fma (/ y a) t x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z / (z - a)), y, x);
	double tmp;
	if (z <= -1850000000.0) {
		tmp = t_1;
	} else if (z <= 2.15e-49) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z / Float64(z - a)), y, x)
	tmp = 0.0
	if (z <= -1850000000.0)
		tmp = t_1;
	elseif (z <= 2.15e-49)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[z, -1850000000.0], t$95$1, If[LessEqual[z, 2.15e-49], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.85e9 or 2.15000000000000008e-49 < z

   1. Initial program 82.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 85.4

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

   5. Applied rewrites 85.4%

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

   if -1.85e9 < z < 2.15000000000000008e-49

   1. Initial program 93.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 81.5

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

   5. Applied rewrites 81.5%

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

Alternative 8: 79.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y}{z - a}, x\right)\\ \mathbf{if}\;z \leq -1850000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ y (- z a)) x)))
   (if (<= z -1850000000.0) t_1 (if (<= z 2.15e-49) (fma (/ y a) t x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, (y / (z - a)), x);
	double tmp;
	if (z <= -1850000000.0) {
		tmp = t_1;
	} else if (z <= 2.15e-49) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(y / Float64(z - a)), x)
	tmp = 0.0
	if (z <= -1850000000.0)
		tmp = t_1;
	elseif (z <= 2.15e-49)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1850000000.0], t$95$1, If[LessEqual[z, 2.15e-49], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.85e9 or 2.15000000000000008e-49 < z

   1. Initial program 82.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 85.4

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

   5. Applied rewrites 85.4%

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

   7. Applied rewrites 83.1%

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

   if -1.85e9 < z < 2.15000000000000008e-49

   1. Initial program 93.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 81.5

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

   5. Applied rewrites 81.5%

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

Alternative 9: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{y}{\frac{z - a}{z - t}} + x \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = (y / ((z - a) / (z - t))) + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 98.0

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

4. Applied rewrites 98.0%

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

5. Final simplification 98.0%

   \[\leadsto \frac{y}{\frac{z - a}{z - t}} + x \]
6. Add Preprocessing

Alternative 10: 76.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+53}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 160:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.3e+53) (+ y x) (if (<= z 160.0) (fma (/ y a) t x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e+53) {
		tmp = y + x;
	} else if (z <= 160.0) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.3e+53)
		tmp = Float64(y + x);
	elseif (z <= 160.0)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.3e+53], N[(y + x), $MachinePrecision], If[LessEqual[z, 160.0], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.3000000000000002e53 or 160 < z

   1. Initial program 78.5%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 81.3

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 81.3%

      \[\leadsto \color{blue}{y + x} \]

   if -2.3000000000000002e53 < z < 160

   1. Initial program 94.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 79.0

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

   5. Applied rewrites 79.0%

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

Alternative 11: 59.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+154}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{a} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y 3.8e+154) (+ y x) (* (/ t a) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= 3.8e+154) {
		tmp = y + x;
	} else {
		tmp = (t / a) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (y <= 3.8d+154) then
        tmp = y + x
    else
        tmp = (t / a) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= 3.8e+154) {
		tmp = y + x;
	} else {
		tmp = (t / a) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= 3.8e+154:
		tmp = y + x
	else:
		tmp = (t / a) * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= 3.8e+154)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(t / a) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= 3.8e+154)
		tmp = y + x;
	else
		tmp = (t / a) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, 3.8e+154], N[(y + x), $MachinePrecision], N[(N[(t / a), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.7999999999999998e154

   1. Initial program 91.1%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 66.6

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{y + x} \]

   if 3.7999999999999998e154 < y

   1. Initial program 67.9%

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

   3. Taylor expanded in y around inf

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

      1. distribute-lft-out-- N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. distribute-rgt-out-- N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 41.8%

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

   10. Applied rewrites 46.3%

       \[\leadsto \frac{t}{a} \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 60.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ y + x \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ y x))

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = y + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{x + y} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{y + x} \]

   2. lower-+.f64 59.6

      \[\leadsto \color{blue}{y + x} \]

5. Applied rewrites 59.6%

   \[\leadsto \color{blue}{y + x} \]
6. Add Preprocessing

Developer Target 1: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ x + \frac{y}{\frac{z - a}{z - t}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = x + (y / ((z - a) / (z - t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024276 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

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

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