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

Percentage Accurate: 85.6% → 98.1%

Time: 6.2s

Alternatives: 11

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: 85.6% 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: 98.1% 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 87.8%

   \[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
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 97.9

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

4. Applied rewrites 97.9%

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

Alternative 2: 76.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+93}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-y}{z}, t, x\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.8e+93)
   (+ t x)
   (if (<= z -1.4e-30)
     (fma (/ (- y) z) t x)
     (if (<= z 3.6e+52) (fma (/ t a) y x) (+ t x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.8e+93) {
		tmp = t + x;
	} else if (z <= -1.4e-30) {
		tmp = fma((-y / z), t, x);
	} else if (z <= 3.6e+52) {
		tmp = fma((t / a), y, x);
	} else {
		tmp = t + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.8e+93)
		tmp = Float64(t + x);
	elseif (z <= -1.4e-30)
		tmp = fma(Float64(Float64(-y) / z), t, x);
	elseif (z <= 3.6e+52)
		tmp = fma(Float64(t / a), y, x);
	else
		tmp = Float64(t + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.8e+93], N[(t + x), $MachinePrecision], If[LessEqual[z, -1.4e-30], N[(N[((-y) / z), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[z, 3.6e+52], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], N[(t + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.79999999999999989e93 or 3.6e52 < z

   1. Initial program 72.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 82.3

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

   5. Applied rewrites 82.3%

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

   if -2.79999999999999989e93 < z < -1.39999999999999994e-30

   1. Initial program 99.6%

      \[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
   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 a around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 76.9

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

   7. Applied rewrites 76.9%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 77.3%

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

   if -1.39999999999999994e-30 < z < 3.6e52

   1. Initial program 96.5%

      \[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
   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 96.2

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 77.0

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

   7. Applied rewrites 77.0%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+93}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-y}{z}, t, x\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+33} \lor \neg \left(z \leq 2.2 \cdot 10^{+52}\right):\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{y}{z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t \cdot y}{a - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -9.5e+33) (not (<= z 2.2e+52)))
   (fma (- 1.0 (/ y z)) t x)
   (+ x (/ (* t y) (- a z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -9.5e+33) || !(z <= 2.2e+52)) {
		tmp = fma((1.0 - (y / z)), t, x);
	} else {
		tmp = x + ((t * y) / (a - z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -9.5e+33) || !(z <= 2.2e+52))
		tmp = fma(Float64(1.0 - Float64(y / z)), t, x);
	else
		tmp = Float64(x + Float64(Float64(t * y) / Float64(a - z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -9.5e+33], N[Not[LessEqual[z, 2.2e+52]], $MachinePrecision]], N[(N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision], N[(x + N[(N[(t * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.5000000000000003e33 or 2.2e52 < z

   1. Initial program 75.6%

      \[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
   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 a around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 91.9

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

   7. Applied rewrites 91.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 91.9%

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

   if -9.5000000000000003e33 < z < 2.2e52

   1. Initial program 96.7%

      \[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
   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 86.9

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

   5. Applied rewrites 86.9%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+33} \lor \neg \left(z \leq 2.2 \cdot 10^{+52}\right):\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{y}{z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t \cdot y}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.7 \cdot 10^{+29} \lor \neg \left(a \leq 4.2 \cdot 10^{-42}\right):\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{y}{z}, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.7e+29) (not (<= a 4.2e-42)))
   (fma (- y z) (/ t a) x)
   (fma (- 1.0 (/ y z)) t x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.7e+29) || !(a <= 4.2e-42)) {
		tmp = fma((y - z), (t / a), x);
	} else {
		tmp = fma((1.0 - (y / z)), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.7e+29) || !(a <= 4.2e-42))
		tmp = fma(Float64(y - z), Float64(t / a), x);
	else
		tmp = fma(Float64(1.0 - Float64(y / z)), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.7e+29], N[Not[LessEqual[a, 4.2e-42]], $MachinePrecision]], N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.69999999999999991e29 or 4.20000000000000013e-42 < a

   1. Initial program 87.5%

      \[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
   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 87.5

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

   5. Applied rewrites 87.5%

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

   if -1.69999999999999991e29 < a < 4.20000000000000013e-42

   1. Initial program 88.1%

      \[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
   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 96.9

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 88.9

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

   7. Applied rewrites 88.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 88.9%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.7 \cdot 10^{+29} \lor \neg \left(a \leq 4.2 \cdot 10^{-42}\right):\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{y}{z}, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+31} \lor \neg \left(z \leq 3.9 \cdot 10^{+52}\right):\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.2e+31) (not (<= z 3.9e+52)))
   (+ t x)
   (fma (- y z) (/ t a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.2e+31) || !(z <= 3.9e+52)) {
		tmp = t + x;
	} else {
		tmp = fma((y - z), (t / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.2e+31) || !(z <= 3.9e+52))
		tmp = Float64(t + x);
	else
		tmp = fma(Float64(y - z), Float64(t / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.2e+31], N[Not[LessEqual[z, 3.9e+52]], $MachinePrecision]], N[(t + x), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.19999999999999991e31 or 3.9e52 < z

   1. Initial program 75.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 79.9

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

   5. Applied rewrites 79.9%

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

   if -1.19999999999999991e31 < z < 3.9e52

   1. Initial program 96.7%

      \[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
   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 79.3

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

   5. Applied rewrites 79.3%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+31} \lor \neg \left(z \leq 3.9 \cdot 10^{+52}\right):\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+31}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, a + z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.2e+31)
   (+ t x)
   (if (<= z 1.45e+114) (fma (- y z) (/ t a) x) (fma (/ t z) (+ a z) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.2e+31) {
		tmp = t + x;
	} else if (z <= 1.45e+114) {
		tmp = fma((y - z), (t / a), x);
	} else {
		tmp = fma((t / z), (a + z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.2e+31)
		tmp = Float64(t + x);
	elseif (z <= 1.45e+114)
		tmp = fma(Float64(y - z), Float64(t / a), x);
	else
		tmp = fma(Float64(t / z), Float64(a + z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.2e+31], N[(t + x), $MachinePrecision], If[LessEqual[z, 1.45e+114], N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(t / z), $MachinePrecision] * N[(a + z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.19999999999999991e31

   1. Initial program 75.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 86.2

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

   5. Applied rewrites 86.2%

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

   if -1.19999999999999991e31 < z < 1.45e114

   1. Initial program 95.9%

      \[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
   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 77.8

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

   5. Applied rewrites 77.8%

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

   if 1.45e114 < z

   1. Initial program 72.8%

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

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower-+.f64 61.3

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

   4. Applied rewrites 61.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 77.2

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

   7. Applied rewrites 77.2%

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

Alternative 7: 76.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+29} \lor \neg \left(z \leq 3.6 \cdot 10^{+52}\right):\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.6e+29) (not (<= z 3.6e+52))) (+ t x) (fma (/ t a) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.6e+29) || !(z <= 3.6e+52)) {
		tmp = t + x;
	} else {
		tmp = fma((t / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.6e+29) || !(z <= 3.6e+52))
		tmp = Float64(t + x);
	else
		tmp = fma(Float64(t / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.6e+29], N[Not[LessEqual[z, 3.6e+52]], $MachinePrecision]], N[(t + x), $MachinePrecision], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.6000000000000002e29 or 3.6e52 < z

   1. Initial program 75.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 79.9

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

   5. Applied rewrites 79.9%

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

   if -4.6000000000000002e29 < z < 3.6e52

   1. Initial program 96.7%

      \[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
   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 96.5

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 75.1

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

   7. Applied rewrites 75.1%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+29} \lor \neg \left(z \leq 3.6 \cdot 10^{+52}\right):\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+29} \lor \neg \left(z \leq 3.6 \cdot 10^{+52}\right):\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.6e+29) (not (<= z 3.6e+52))) (+ t x) (fma (/ y a) t x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.6e+29) || !(z <= 3.6e+52)) {
		tmp = t + x;
	} else {
		tmp = fma((y / a), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.6e+29) || !(z <= 3.6e+52))
		tmp = Float64(t + x);
	else
		tmp = fma(Float64(y / a), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.6e+29], N[Not[LessEqual[z, 3.6e+52]], $MachinePrecision]], N[(t + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.6000000000000002e29 or 3.6e52 < z

   1. Initial program 75.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 79.9

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

   5. Applied rewrites 79.9%

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

   if -4.6000000000000002e29 < z < 3.6e52

   1. Initial program 96.7%

      \[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
   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 74.3

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

   5. Applied rewrites 74.3%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+29} \lor \neg \left(z \leq 3.6 \cdot 10^{+52}\right):\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+180}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+250}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{x}, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y 7.2e+180)
   (+ t x)
   (if (<= y 1.4e+250) (* (/ y a) t) (fma (/ t x) x x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= 7.2e+180) {
		tmp = t + x;
	} else if (y <= 1.4e+250) {
		tmp = (y / a) * t;
	} else {
		tmp = fma((t / x), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= 7.2e+180)
		tmp = Float64(t + x);
	elseif (y <= 1.4e+250)
		tmp = Float64(Float64(y / a) * t);
	else
		tmp = fma(Float64(t / x), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, 7.2e+180], N[(t + x), $MachinePrecision], If[LessEqual[y, 1.4e+250], N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision], N[(N[(t / x), $MachinePrecision] * x + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.2000000000000004e180

   1. Initial program 87.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 64.4

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

   5. Applied rewrites 64.4%

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

   if 7.2000000000000004e180 < y < 1.40000000000000005e250

   1. Initial program 93.1%

      \[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
   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 86.1

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

   5. Applied rewrites 86.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 73.0%

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

   10. Applied rewrites 73.0%

       \[\leadsto \frac{y}{a} \cdot t \]

   if 1.40000000000000005e250 < y

   1. Initial program 86.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 44.8

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 65.2%

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

Alternative 10: 59.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+180} \lor \neg \left(y \leq 9.5 \cdot 10^{+261}\right):\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= y 7.2e+180) (not (<= y 9.5e+261))) (+ t x) (* (/ y a) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= 7.2e+180) || !(y <= 9.5e+261)) {
		tmp = t + x;
	} else {
		tmp = (y / a) * t;
	}
	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 <= 7.2d+180) .or. (.not. (y <= 9.5d+261))) then
        tmp = t + x
    else
        tmp = (y / a) * t
    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 <= 7.2e+180) || !(y <= 9.5e+261)) {
		tmp = t + x;
	} else {
		tmp = (y / a) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (y <= 7.2e+180) or not (y <= 9.5e+261):
		tmp = t + x
	else:
		tmp = (y / a) * t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((y <= 7.2e+180) || !(y <= 9.5e+261))
		tmp = Float64(t + x);
	else
		tmp = Float64(Float64(y / a) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((y <= 7.2e+180) || ~((y <= 9.5e+261)))
		tmp = t + x;
	else
		tmp = (y / a) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[y, 7.2e+180], N[Not[LessEqual[y, 9.5e+261]], $MachinePrecision]], N[(t + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.2000000000000004e180 or 9.50000000000000085e261 < y

   1. Initial program 87.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 63.8

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

   5. Applied rewrites 63.8%

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

   if 7.2000000000000004e180 < y < 9.50000000000000085e261

   1. Initial program 93.9%

      \[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
   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 87.8

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

   5. Applied rewrites 87.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 76.4%

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

   10. Applied rewrites 76.4%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+180} \lor \neg \left(y \leq 9.5 \cdot 10^{+261}\right):\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.7% 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 87.8%

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

3. Taylor expanded in z around inf

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

   1. lower-+.f64 60.6

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

5. Applied rewrites 60.6%

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

Developer Target 1: 99.2% 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 2024339 
(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))))