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

Percentage Accurate: 85.0% → 98.2%

Time: 9.8s

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: 85.0% 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: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-t, \left(y - z\right) \cdot \frac{1}{z - a}, x\right) \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (fma (- t) (* (- y z) (/ 1.0 (- z a))) x))

C

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

Julia

function code(x, y, z, t, a)
	return fma(Float64(-t), Float64(Float64(y - z) * Float64(1.0 / Float64(z - a))), x)
end

Wolfram

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

Derivation

1. Initial program 86.2%

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

   1. +-commutative N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. *-commutative N/A

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

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

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

   6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   12. neg-sub0 N/A

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

   13. sub-neg N/A

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

   14. +-commutative N/A

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

   15. associate--r+ N/A

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

   16. neg-sub0 N/A

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

   17. remove-double-neg N/A

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

   18. --lowering--.f64 99.4

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

4. Applied egg-rr 99.4%

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

Alternative 2: 87.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.48 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, t, x\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a - z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, 1 - \frac{y}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.48e-5)
   (fma (/ z (- z a)) t x)
   (if (<= z 2e+52) (fma (/ t (- a z)) y x) (fma t (- 1.0 (/ y z)) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.48e-5) {
		tmp = fma((z / (z - a)), t, x);
	} else if (z <= 2e+52) {
		tmp = fma((t / (a - z)), y, x);
	} else {
		tmp = fma(t, (1.0 - (y / z)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.48e-5)
		tmp = fma(Float64(z / Float64(z - a)), t, x);
	elseif (z <= 2e+52)
		tmp = fma(Float64(t / Float64(a - z)), y, x);
	else
		tmp = fma(t, Float64(1.0 - Float64(y / z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.48e-5], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[z, 2e+52], N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(t * N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.4800000000000001e-5

   1. Initial program 77.7%

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

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. *-commutative N/A

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

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

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      12. neg-sub0 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. associate--r+ N/A

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

      16. neg-sub0 N/A

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

      17. remove-double-neg N/A

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

      18. --lowering--.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. --lowering--.f64 89.8

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

   7. Simplified 89.8%

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

   if -1.4800000000000001e-5 < z < 2e52

   1. Initial program 95.2%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 96.5

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

   4. Applied egg-rr 96.5%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 89.6%

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

   if 2e52 < z

   1. Initial program 72.8%

      \[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
   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. distribute-rgt-neg-in N/A

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

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

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

      6. neg-sub0 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. --lowering--.f64 N/A

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

      16. /-lowering-/.f64 94.0

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

   5. Simplified 94.0%

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

Alternative 3: 88.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, 1 - \frac{y}{z}, x\right)\\ \mathbf{if}\;z \leq -2.3 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a - z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma t (- 1.0 (/ y z)) x)))
   (if (<= z -2.3e+21) t_1 (if (<= z 2.6e+51) (fma (/ t (- a z)) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(t, (1.0 - (y / z)), x);
	double tmp;
	if (z <= -2.3e+21) {
		tmp = t_1;
	} else if (z <= 2.6e+51) {
		tmp = fma((t / (a - z)), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(t, Float64(1.0 - Float64(y / z)), x)
	tmp = 0.0
	if (z <= -2.3e+21)
		tmp = t_1;
	elseif (z <= 2.6e+51)
		tmp = fma(Float64(t / Float64(a - z)), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -2.3e+21], t$95$1, If[LessEqual[z, 2.6e+51], N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.3e21 or 2.6000000000000001e51 < z

   1. Initial program 74.8%

      \[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
   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. distribute-rgt-neg-in N/A

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

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

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

      6. neg-sub0 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. --lowering--.f64 N/A

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

      16. /-lowering-/.f64 88.6

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

   5. Simplified 88.6%

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

   if -2.3e21 < z < 2.6000000000000001e51

   1. Initial program 95.3%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 96.6

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

   4. Applied egg-rr 96.6%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 89.2%

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

Alternative 4: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \frac{y - z}{a}, x\right)\\ \mathbf{if}\;a \leq -2.35 \cdot 10^{+64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 65000000000:\\ \;\;\;\;\mathsf{fma}\left(t, 1 - \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma t (/ (- y z) a) x)))
   (if (<= a -2.35e+64)
     t_1
     (if (<= a 65000000000.0) (fma t (- 1.0 (/ y z)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(t, ((y - z) / a), x);
	double tmp;
	if (a <= -2.35e+64) {
		tmp = t_1;
	} else if (a <= 65000000000.0) {
		tmp = fma(t, (1.0 - (y / z)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(t, Float64(Float64(y - z) / a), x)
	tmp = 0.0
	if (a <= -2.35e+64)
		tmp = t_1;
	elseif (a <= 65000000000.0)
		tmp = fma(t, Float64(1.0 - Float64(y / z)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -2.35e+64], t$95$1, If[LessEqual[a, 65000000000.0], N[(t * N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.35000000000000015e64 or 6.5e10 < a

   1. Initial program 84.0%

      \[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
   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. associate-/l* N/A

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

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

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

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

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

      5. --lowering--.f64 92.1

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

   5. Simplified 92.1%

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

   if -2.35000000000000015e64 < a < 6.5e10

   1. Initial program 88.0%

      \[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
   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. distribute-rgt-neg-in N/A

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

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

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

      6. neg-sub0 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. --lowering--.f64 N/A

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

      16. /-lowering-/.f64 84.0

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

   5. Simplified 84.0%

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

Alternative 5: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{if}\;a \leq -1.2 \cdot 10^{+67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 820000000000:\\ \;\;\;\;\mathsf{fma}\left(t, 1 - \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ t a) x)))
   (if (<= a -1.2e+67)
     t_1
     (if (<= a 820000000000.0) (fma t (- 1.0 (/ y z)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (t / a), x);
	double tmp;
	if (a <= -1.2e+67) {
		tmp = t_1;
	} else if (a <= 820000000000.0) {
		tmp = fma(t, (1.0 - (y / z)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(t / a), x)
	tmp = 0.0
	if (a <= -1.2e+67)
		tmp = t_1;
	elseif (a <= 820000000000.0)
		tmp = fma(t, Float64(1.0 - Float64(y / z)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.2e+67], t$95$1, If[LessEqual[a, 820000000000.0], N[(t * N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.20000000000000001e67 or 8.2e11 < a

   1. Initial program 84.0%

      \[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
   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. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 82.1

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

   5. Simplified 82.1%

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

   if -1.20000000000000001e67 < a < 8.2e11

   1. Initial program 88.0%

      \[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
   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. distribute-rgt-neg-in N/A

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

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

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

      6. neg-sub0 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. --lowering--.f64 N/A

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

      16. /-lowering-/.f64 84.0

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

   5. Simplified 84.0%

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

Alternative 6: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+16}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;z \leq 2700000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.2e+16) (+ t x) (if (<= z 2700000.0) (fma (/ y a) t x) (+ t x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.2e+16) {
		tmp = t + x;
	} else if (z <= 2700000.0) {
		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 <= -3.2e+16)
		tmp = Float64(t + x);
	elseif (z <= 2700000.0)
		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, -3.2e+16], N[(t + x), $MachinePrecision], If[LessEqual[z, 2700000.0], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(t + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.2e16 or 2.7e6 < z

   1. Initial program 75.9%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 79.4%

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

   if -3.2e16 < z < 2.7e6

   1. Initial program 95.7%

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

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

      2. *-lowering-*.f64 75.8

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

   5. Simplified 75.8%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. /-lowering-/.f64 78.5

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

   7. Applied egg-rr 78.5%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+16}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;z \leq 2700000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+18}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;z \leq 50000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.2e+18) (+ t x) (if (<= z 50000.0) (fma y (/ t a) x) (+ t x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.2e+18) {
		tmp = t + x;
	} else if (z <= 50000.0) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = t + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.2e+18)
		tmp = Float64(t + x);
	elseif (z <= 50000.0)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = Float64(t + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.2e+18], N[(t + x), $MachinePrecision], If[LessEqual[z, 50000.0], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(t + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.2e18 or 5e4 < z

   1. Initial program 75.9%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 79.4%

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

   if -7.2e18 < z < 5e4

   1. Initial program 95.7%

      \[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
   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. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 77.8

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

   5. Simplified 77.8%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+18}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;z \leq 50000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. --lowering--.f64 96.9

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

4. Applied egg-rr 96.9%

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

Alternative 9: 63.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.35 \cdot 10^{+113}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 5.3 \cdot 10^{+212}:\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.35e+113) x (if (<= a 5.3e+212) (+ t x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.35e+113) {
		tmp = x;
	} else if (a <= 5.3e+212) {
		tmp = t + x;
	} else {
		tmp = 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 <= (-2.35d+113)) then
        tmp = x
    else if (a <= 5.3d+212) then
        tmp = t + x
    else
        tmp = 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 <= -2.35e+113) {
		tmp = x;
	} else if (a <= 5.3e+212) {
		tmp = t + x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -2.35e+113:
		tmp = x
	elif a <= 5.3e+212:
		tmp = t + x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.35e+113)
		tmp = x;
	elseif (a <= 5.3e+212)
		tmp = Float64(t + x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -2.35e+113)
		tmp = x;
	elseif (a <= 5.3e+212)
		tmp = t + x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.35e+113], x, If[LessEqual[a, 5.3e+212], N[(t + x), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -2.3499999999999999e113 or 5.29999999999999993e212 < a

   1. Initial program 82.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 67.0%

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

   if -2.3499999999999999e113 < a < 5.29999999999999993e212

   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

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

   5. Simplified 64.8%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.35 \cdot 10^{+113}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 5.3 \cdot 10^{+212}:\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-235}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-152}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -3.8e-235) x (if (<= x 1.4e-152) t x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -3.8e-235) {
		tmp = x;
	} else if (x <= 1.4e-152) {
		tmp = t;
	} else {
		tmp = 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 (x <= (-3.8d-235)) then
        tmp = x
    else if (x <= 1.4d-152) then
        tmp = t
    else
        tmp = 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 (x <= -3.8e-235) {
		tmp = x;
	} else if (x <= 1.4e-152) {
		tmp = t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -3.8e-235:
		tmp = x
	elif x <= 1.4e-152:
		tmp = t
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -3.8e-235)
		tmp = x;
	elseif (x <= 1.4e-152)
		tmp = t;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -3.8e-235)
		tmp = x;
	elseif (x <= 1.4e-152)
		tmp = t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -3.8e-235], x, If[LessEqual[x, 1.4e-152], t, x]]

Derivation

1. Split input into 2 regimes

2. if x < -3.80000000000000026e-235 or 1.39999999999999992e-152 < x

   1. Initial program 85.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 59.8%

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

   if -3.80000000000000026e-235 < x < 1.39999999999999992e-152

   1. Initial program 89.5%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 46.0%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 41.2%

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

Alternative 11: 18.6% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := t

Derivation

1. Initial program 86.2%

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

3. Taylor expanded in z around inf

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

5. Simplified 60.7%

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

6. Taylor expanded in x around 0

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

8. Simplified 17.4%

   \[\leadsto \color{blue}{t} \]
9. 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 2024204 
(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))))