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

Percentage Accurate: 85.6% → 98.3%

Time: 30.0s

Alternatives: 7

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 98.4

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

4. Applied rewrites 98.4%

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

Alternative 2: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \mathbf{elif}\;z \leq 0.0065:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.8e-24)
   (fma y (- 1.0 (/ t z)) x)
   (if (<= z 0.0065) (fma y (/ (- t z) a) x) (fma y (/ z (- z a)) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.8e-24) {
		tmp = fma(y, (1.0 - (t / z)), x);
	} else if (z <= 0.0065) {
		tmp = fma(y, ((t - z) / a), x);
	} else {
		tmp = fma(y, (z / (z - a)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.8e-24)
		tmp = fma(y, Float64(1.0 - Float64(t / z)), x);
	elseif (z <= 0.0065)
		tmp = fma(y, Float64(Float64(t - z) / a), x);
	else
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.8e-24], N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 0.0065], N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.7999999999999997e-24

   1. Initial program 73.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 92.0

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

   5. Applied rewrites 92.0%

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

   if -5.7999999999999997e-24 < z < 0.0064999999999999997

   1. Initial program 93.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-neg N/A

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

      4. lift--.f64 N/A

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

      5. remove-double-neg N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 97.6

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

   4. Applied rewrites 97.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 97.6

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

   6. Applied rewrites 97.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 97.6

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

      4. remove-double-neg N/A

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

      5. lift-neg.f64 N/A

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

      6. lower-neg.f64 97.6

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 97.6

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

   8. Applied rewrites 97.6%

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

   9. Taylor expanded in a around inf

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

       1. +-commutative N/A

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

       2. associate-/l* N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower--.f64 82.5

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

   11. Applied rewrites 82.5%

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

   if 0.0064999999999999997 < z

   1. Initial program 68.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-neg N/A

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

      4. lift--.f64 N/A

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

      5. remove-double-neg N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 88.1

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

   7. Applied rewrites 88.1%

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

Alternative 3: 82.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4e-29)
   (fma y (- 1.0 (/ t z)) x)
   (if (<= z 8.4e-14) (fma t (/ y a) x) (fma y (/ z (- z a)) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4e-29) {
		tmp = fma(y, (1.0 - (t / z)), x);
	} else if (z <= 8.4e-14) {
		tmp = fma(t, (y / a), x);
	} else {
		tmp = fma(y, (z / (z - a)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4e-29)
		tmp = fma(y, Float64(1.0 - Float64(t / z)), x);
	elseif (z <= 8.4e-14)
		tmp = fma(t, Float64(y / a), x);
	else
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4e-29], N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 8.4e-14], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.99999999999999977e-29

   1. Initial program 73.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 90.7

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

   5. Applied rewrites 90.7%

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

   if -3.99999999999999977e-29 < z < 8.3999999999999995e-14

   1. Initial program 93.5%

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

      1. flip-- N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. frac-2neg N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. difference-of-squares N/A

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

      11. lift--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-+.f64 76.6

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

   4. Applied rewrites 76.6%

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

   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

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 80.1

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

   7. Applied rewrites 80.1%

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

   if 8.3999999999999995e-14 < z

   1. Initial program 69.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-neg N/A

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

      4. lift--.f64 N/A

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

      5. remove-double-neg N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 87.1

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

   7. Applied rewrites 87.1%

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

Alternative 4: 82.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \mathbf{if}\;z \leq -4 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.012:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (- 1.0 (/ t z)) x)))
   (if (<= z -4e-29) t_1 (if (<= z 0.012) (fma t (/ y a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (1.0 - (t / z)), x);
	double tmp;
	if (z <= -4e-29) {
		tmp = t_1;
	} else if (z <= 0.012) {
		tmp = fma(t, (y / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(1.0 - Float64(t / z)), x)
	tmp = 0.0
	if (z <= -4e-29)
		tmp = t_1;
	elseif (z <= 0.012)
		tmp = fma(t, Float64(y / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -4e-29], t$95$1, If[LessEqual[z, 0.012], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.99999999999999977e-29 or 0.012 < z

   1. Initial program 71.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 88.0

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

   5. Applied rewrites 88.0%

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

   if -3.99999999999999977e-29 < z < 0.012

   1. Initial program 93.7%

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

      1. flip-- N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. frac-2neg N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. difference-of-squares N/A

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

      11. lift--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-+.f64 76.4

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

   4. Applied rewrites 76.4%

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

   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

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 79.0

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

   7. Applied rewrites 79.0%

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

Alternative 5: 77.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{-29}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 0.052:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.5e-29) (+ y x) (if (<= z 0.052) (fma t (/ y a) x) (+ y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.5e-29) {
		tmp = y + x;
	} else if (z <= 0.052) {
		tmp = fma(t, (y / a), x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.5e-29)
		tmp = Float64(y + x);
	elseif (z <= 0.052)
		tmp = fma(t, Float64(y / a), x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.5e-29], N[(y + x), $MachinePrecision], If[LessEqual[z, 0.052], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.4999999999999998e-29 or 0.0519999999999999976 < z

   1. Initial program 71.1%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -4.4999999999999998e-29 < z < 0.0519999999999999976

   1. Initial program 93.7%

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

      1. flip-- N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. frac-2neg N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. difference-of-squares N/A

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

      11. lift--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-+.f64 76.4

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

   4. Applied rewrites 76.4%

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

   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

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 79.0

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

   7. Applied rewrites 79.0%

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

Alternative 6: 96.1% 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(y / Float64(z - a)), Float64(z - t), x)
end

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-/l* N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 95.4

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

4. Applied rewrites 95.4%

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

Alternative 7: 59.8% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

3. Taylor expanded in z around inf

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

   1. +-commutative N/A

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

   2. lower-+.f64 62.3

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

5. Applied rewrites 62.3%

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

Developer Target 1: 98.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024220 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

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

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