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

Percentage Accurate: 85.2% → 98.4%

Time: 7.3s

Alternatives: 12

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

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) / (a - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) / (a - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t \cdot \left(y - z\right)}{a - z} + x \leq 4 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{\frac{a - z}{t}} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (+ (/ (* t (- y z)) (- a z)) x) 4e-32)
   (fma (/ (- y z) (- a 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 ((((t * (y - z)) / (a - z)) + x) <= 4e-32) {
		tmp = fma(((y - z) / (a - 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(t * Float64(y - z)) / Float64(a - z)) + x) <= 4e-32)
		tmp = fma(Float64(Float64(y - z) / Float64(a - z)), t, x);
	else
		tmp = Float64(Float64(Float64(y - z) / Float64(Float64(a - z) / t)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], 4e-32], N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] / N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))) < 4.00000000000000022e-32

   1. Initial program 84.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 98.7

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

   4. Applied rewrites 98.7%

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

   if 4.00000000000000022e-32 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)))

   1. Initial program 82.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot \left(y - z\right)}{a - z} + x \leq 4 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{\frac{a - z}{t}} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t \cdot \left(y - z\right)}{a - z}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+40}:\\ \;\;\;\;\frac{t}{a - z} \cdot \left(y - z\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - z}, -t, x\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+273}:\\ \;\;\;\;\frac{t \cdot y}{a - z} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{a - z} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* t (- y z)) (- a z))))
   (if (<= t_1 -4e+40)
     (* (/ t (- a z)) (- y z))
     (if (<= t_1 5e-53)
       (fma (/ z (- a z)) (- t) x)
       (if (<= t_1 4e+273)
         (+ (/ (* t y) (- a z)) x)
         (* (/ (- y z) (- a z)) t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t * (y - z)) / (a - z);
	double tmp;
	if (t_1 <= -4e+40) {
		tmp = (t / (a - z)) * (y - z);
	} else if (t_1 <= 5e-53) {
		tmp = fma((z / (a - z)), -t, x);
	} else if (t_1 <= 4e+273) {
		tmp = ((t * y) / (a - z)) + x;
	} else {
		tmp = ((y - z) / (a - z)) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t * Float64(y - z)) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= -4e+40)
		tmp = Float64(Float64(t / Float64(a - z)) * Float64(y - z));
	elseif (t_1 <= 5e-53)
		tmp = fma(Float64(z / Float64(a - z)), Float64(-t), x);
	elseif (t_1 <= 4e+273)
		tmp = Float64(Float64(Float64(t * y) / Float64(a - z)) + x);
	else
		tmp = Float64(Float64(Float64(y - z) / Float64(a - z)) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+40], N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-53], N[(N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision] * (-t) + x), $MachinePrecision], If[LessEqual[t$95$1, 4e+273], N[(N[(N[(t * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < -4.00000000000000012e40

   1. Initial program 56.6%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

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

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 88.8

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

   5. Applied rewrites 88.8%

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

   if -4.00000000000000012e40 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < 5e-53

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 92.8

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

   5. Applied rewrites 92.8%

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

   if 5e-53 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < 3.99999999999999978e273

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 86.1

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

   5. Applied rewrites 86.1%

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

   if 3.99999999999999978e273 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))

   1. Initial program 44.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

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

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 91.5

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

   5. Applied rewrites 91.5%

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

   7. Applied rewrites 91.5%

      \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{t} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot \left(y - z\right)}{a - z} \leq -4 \cdot 10^{+40}:\\ \;\;\;\;\frac{t}{a - z} \cdot \left(y - z\right)\\ \mathbf{elif}\;\frac{t \cdot \left(y - z\right)}{a - z} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - z}, -t, x\right)\\ \mathbf{elif}\;\frac{t \cdot \left(y - z\right)}{a - z} \leq 4 \cdot 10^{+273}:\\ \;\;\;\;\frac{t \cdot y}{a - z} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{a - z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{a - z} \cdot \left(y - z\right)\\ t_2 := \frac{t \cdot \left(y - z\right)}{a - z}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - z}, -t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ t (- a z)) (- y z))) (t_2 (/ (* t (- y z)) (- a z))))
   (if (<= t_2 -4e+40)
     t_1
     (if (<= t_2 2e+94) (fma (/ z (- a z)) (- t) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t / (a - z)) * (y - z);
	double t_2 = (t * (y - z)) / (a - z);
	double tmp;
	if (t_2 <= -4e+40) {
		tmp = t_1;
	} else if (t_2 <= 2e+94) {
		tmp = fma((z / (a - z)), -t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t / Float64(a - z)) * Float64(y - z))
	t_2 = Float64(Float64(t * Float64(y - z)) / Float64(a - z))
	tmp = 0.0
	if (t_2 <= -4e+40)
		tmp = t_1;
	elseif (t_2 <= 2e+94)
		tmp = fma(Float64(z / Float64(a - z)), Float64(-t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+40], t$95$1, If[LessEqual[t$95$2, 2e+94], N[(N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision] * (-t) + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < -4.00000000000000012e40 or 2e94 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))

   1. Initial program 57.4%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

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

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 88.0

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

   5. Applied rewrites 88.0%

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

   if -4.00000000000000012e40 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < 2e94

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 87.8

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

   5. Applied rewrites 87.8%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot \left(y - z\right)}{a - z} \leq -4 \cdot 10^{+40}:\\ \;\;\;\;\frac{t}{a - z} \cdot \left(y - z\right)\\ \mathbf{elif}\;\frac{t \cdot \left(y - z\right)}{a - z} \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - z}, -t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{a - z} \cdot \left(y - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t, x\right)\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{-98}:\\ \;\;\;\;\mathsf{fma}\left(z - y, \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.1e+82)
   (fma (/ (- y z) a) t x)
   (if (<= a 3.8e-98) (fma (- z y) (/ t z) x) (fma (- y z) (/ t a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.1e+82) {
		tmp = fma(((y - z) / a), t, x);
	} else if (a <= 3.8e-98) {
		tmp = fma((z - y), (t / z), x);
	} else {
		tmp = fma((y - z), (t / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.1e+82)
		tmp = fma(Float64(Float64(y - z) / a), t, x);
	elseif (a <= 3.8e-98)
		tmp = fma(Float64(z - y), Float64(t / z), x);
	else
		tmp = fma(Float64(y - z), Float64(t / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.1e+82], N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[a, 3.8e-98], N[(N[(z - y), $MachinePrecision] * N[(t / z), $MachinePrecision] + x), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.1000000000000001e82

   1. Initial program 77.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 89.5

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

   7. Applied rewrites 89.5%

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

   if -1.1000000000000001e82 < a < 3.8000000000000003e-98

   1. Initial program 84.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 85.4

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

   7. Applied rewrites 85.4%

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

   if 3.8000000000000003e-98 < a

   1. Initial program 87.3%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 84.8

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

   5. Applied rewrites 84.8%

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

Alternative 5: 79.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \mathbf{if}\;a \leq -1.1 \cdot 10^{+82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{-98}:\\ \;\;\;\;\mathsf{fma}\left(z - y, \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ t a) x)))
   (if (<= a -1.1e+82) t_1 (if (<= a 3.8e-98) (fma (- z y) (/ t z) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - z), (t / a), x);
	double tmp;
	if (a <= -1.1e+82) {
		tmp = t_1;
	} else if (a <= 3.8e-98) {
		tmp = fma((z - y), (t / z), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - z), Float64(t / a), x)
	tmp = 0.0
	if (a <= -1.1e+82)
		tmp = t_1;
	elseif (a <= 3.8e-98)
		tmp = fma(Float64(z - y), Float64(t / z), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.1e+82], t$95$1, If[LessEqual[a, 3.8e-98], N[(N[(z - y), $MachinePrecision] * N[(t / z), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.1000000000000001e82 or 3.8000000000000003e-98 < a

   1. Initial program 83.5%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 86.7

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

   5. Applied rewrites 86.7%

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

   if -1.1000000000000001e82 < a < 3.8000000000000003e-98

   1. Initial program 84.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 85.4

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

   7. Applied rewrites 85.4%

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

Alternative 6: 81.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \mathbf{if}\;a \leq -1.1 \cdot 10^{+82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{y}{z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ t a) x)))
   (if (<= a -1.1e+82) t_1 (if (<= a 6e-34) (fma (- 1.0 (/ y z)) t x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - z), (t / a), x);
	double tmp;
	if (a <= -1.1e+82) {
		tmp = t_1;
	} else if (a <= 6e-34) {
		tmp = fma((1.0 - (y / z)), t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - z), Float64(t / a), x)
	tmp = 0.0
	if (a <= -1.1e+82)
		tmp = t_1;
	elseif (a <= 6e-34)
		tmp = fma(Float64(1.0 - Float64(y / z)), t, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.1e+82], t$95$1, If[LessEqual[a, 6e-34], N[(N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.1000000000000001e82 or 6e-34 < a

   1. Initial program 81.7%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 87.9

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

   5. Applied rewrites 87.9%

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

   if -1.1000000000000001e82 < a < 6e-34

   1. Initial program 85.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \frac{y - z}{z}}, t, x\right) \]

      8. div-sub N/A

         \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(\frac{y}{z} - \frac{z}{z}\right)}, t, x\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(0 - \left(\frac{y}{z} - \color{blue}{1}\right), t, x\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - \frac{y}{z}\right) + 1}, t, x\right) \]

      11. neg-sub0 N/A

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

      12. mul-1-neg N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 83.4

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

   5. Applied rewrites 83.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - \frac{y}{z}, t, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 - \frac{y}{z}, t, x\right)\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{t \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- 1.0 (/ y z)) t x)))
   (if (<= z -7.5e-55) t_1 (if (<= z 5.6e-14) (+ (/ (* t y) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((1.0 - (y / z)), t, x);
	double tmp;
	if (z <= -7.5e-55) {
		tmp = t_1;
	} else if (z <= 5.6e-14) {
		tmp = ((t * y) / a) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(1.0 - Float64(y / z)), t, x)
	tmp = 0.0
	if (z <= -7.5e-55)
		tmp = t_1;
	elseif (z <= 5.6e-14)
		tmp = Float64(Float64(Float64(t * y) / a) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision]}, If[LessEqual[z, -7.5e-55], t$95$1, If[LessEqual[z, 5.6e-14], N[(N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.50000000000000023e-55 or 5.6000000000000001e-14 < z

   1. Initial program 75.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \frac{y - z}{z}}, t, x\right) \]

      8. div-sub N/A

         \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(\frac{y}{z} - \frac{z}{z}\right)}, t, x\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(0 - \left(\frac{y}{z} - \color{blue}{1}\right), t, x\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - \frac{y}{z}\right) + 1}, t, x\right) \]

      11. neg-sub0 N/A

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

      12. mul-1-neg N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 85.3

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

   5. Applied rewrites 85.3%

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

   if -7.50000000000000023e-55 < z < 5.6000000000000001e-14

   1. Initial program 97.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 82.1

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

   5. Applied rewrites 82.1%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{y}{z}, t, x\right)\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{t \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{y}{z}, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{-40}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{t \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.1e-40) (+ t x) (if (<= z 1.8e-7) (+ (/ (* t y) a) x) (+ t x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.1e-40) {
		tmp = t + x;
	} else if (z <= 1.8e-7) {
		tmp = ((t * y) / a) + x;
	} else {
		tmp = 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 <= (-4.1d-40)) then
        tmp = t + x
    else if (z <= 1.8d-7) then
        tmp = ((t * y) / a) + x
    else
        tmp = 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 <= -4.1e-40) {
		tmp = t + x;
	} else if (z <= 1.8e-7) {
		tmp = ((t * y) / a) + x;
	} else {
		tmp = t + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.1e-40:
		tmp = t + x
	elif z <= 1.8e-7:
		tmp = ((t * y) / a) + x
	else:
		tmp = t + x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.1e-40)
		tmp = Float64(t + x);
	elseif (z <= 1.8e-7)
		tmp = Float64(Float64(Float64(t * y) / a) + x);
	else
		tmp = Float64(t + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4.1e-40)
		tmp = t + x;
	elseif (z <= 1.8e-7)
		tmp = ((t * y) / a) + x;
	else
		tmp = t + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.1e-40], N[(t + x), $MachinePrecision], If[LessEqual[z, 1.8e-7], N[(N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(t + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.09999999999999963e-40 or 1.79999999999999997e-7 < z

   1. Initial program 75.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 78.6

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

   5. Applied rewrites 78.6%

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

   if -4.09999999999999963e-40 < z < 1.79999999999999997e-7

   1. Initial program 97.1%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 80.0

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

   5. Applied rewrites 80.0%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{-40}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{t \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{-40}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.1e-40) (+ t x) (if (<= z 2.3e-7) (fma (/ y a) t x) (+ t x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.1e-40) {
		tmp = t + x;
	} else if (z <= 2.3e-7) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = t + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.1e-40)
		tmp = Float64(t + x);
	elseif (z <= 2.3e-7)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(t + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.1e-40], N[(t + x), $MachinePrecision], If[LessEqual[z, 2.3e-7], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(t + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.09999999999999963e-40 or 2.29999999999999995e-7 < z

   1. Initial program 75.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 78.6

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

   5. Applied rewrites 78.6%

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

   if -4.09999999999999963e-40 < z < 2.29999999999999995e-7

   1. Initial program 97.1%

      \[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
   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 78.0

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

   5. Applied rewrites 78.0%

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

Alternative 10: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 97.3

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

4. Applied rewrites 97.3%

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

Alternative 11: 62.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.3 \cdot 10^{+129}:\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a 5.3e+129) (+ t x) (* -1.0 (- x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= 5.3e+129) {
		tmp = t + x;
	} else {
		tmp = -1.0 * -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 (a <= 5.3d+129) then
        tmp = t + x
    else
        tmp = (-1.0d0) * -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= 5.3e+129) {
		tmp = t + x;
	} else {
		tmp = -1.0 * -x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= 5.3e+129:
		tmp = t + x
	else:
		tmp = -1.0 * -x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= 5.3e+129)
		tmp = Float64(t + x);
	else
		tmp = Float64(-1.0 * Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= 5.3e+129)
		tmp = t + x;
	else
		tmp = -1.0 * -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, 5.3e+129], N[(t + x), $MachinePrecision], N[(-1.0 * (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.2999999999999999e129

   1. Initial program 82.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 64.7

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

   5. Applied rewrites 64.7%

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

   if 5.2999999999999999e129 < a

   1. Initial program 89.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. distribute-lft-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

   7. Applied rewrites 97.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \left(-x\right) \cdot -1 \]
   9. Step-by-step derivation

   10. Applied rewrites 71.4%

       \[\leadsto \left(-x\right) \cdot -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.3 \cdot 10^{+129}:\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.7% accurate, 6.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return 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 = t + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Taylor expanded in z around inf

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

   1. lower-+.f64 63.4

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

5. Applied rewrites 63.4%

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

Developer Target 1: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y - z}{a - z} \cdot t\\ \mathbf{if}\;t < -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (/ (- y z) (- a z)) t))))
   (if (< t -1.0682974490174067e-39)
     t_1
     (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - z) / (a - z)) * t);
	double tmp;
	if (t < -1.0682974490174067e-39) {
		tmp = t_1;
	} else if (t < 3.9110949887586375e-141) {
		tmp = x + (((y - z) * t) / (a - z));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = x + (((y - z) / (a - z)) * t)
    if (t < (-1.0682974490174067d-39)) then
        tmp = t_1
    else if (t < 3.9110949887586375d-141) then
        tmp = x + (((y - z) * t) / (a - z))
    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 t_1 = x + (((y - z) / (a - z)) * t);
	double tmp;
	if (t < -1.0682974490174067e-39) {
		tmp = t_1;
	} else if (t < 3.9110949887586375e-141) {
		tmp = x + (((y - z) * t) / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (((y - z) / (a - z)) * t)
	tmp = 0
	if t < -1.0682974490174067e-39:
		tmp = t_1
	elif t < 3.9110949887586375e-141:
		tmp = x + (((y - z) * t) / (a - z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - z) / Float64(a - z)) * t))
	tmp = 0.0
	if (t < -1.0682974490174067e-39)
		tmp = t_1;
	elseif (t < 3.9110949887586375e-141)
		tmp = Float64(x + Float64(Float64(Float64(y - z) * t) / Float64(a - z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (((y - z) / (a - z)) * t);
	tmp = 0.0;
	if (t < -1.0682974490174067e-39)
		tmp = t_1;
	elseif (t < 3.9110949887586375e-141)
		tmp = x + (((y - z) * t) / (a - z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -1.0682974490174067e-39], t$95$1, If[Less[t, 3.9110949887586375e-141], N[(x + N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024294 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -10682974490174067/10000000000000000000000000000000000000000000000000000000) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 312887599100691/80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t)))))

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