Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B

Percentage Accurate: 98.1% → 98.3%

Time: 7.9s

Alternatives: 14

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 N/A

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

   6. frac-2neg N/A

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

   7. lower-/.f64 N/A

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

   8. neg-sub0 N/A

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. +-commutative N/A

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

   12. associate--r+ N/A

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

   13. neg-sub0 N/A

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

   14. remove-double-neg N/A

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

   15. lower--.f64 N/A

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

   16. neg-sub0 N/A

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

   17. lift--.f64 N/A

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

   18. sub-neg N/A

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

   19. +-commutative N/A

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

   20. associate--r+ N/A

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

   21. neg-sub0 N/A

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

   22. remove-double-neg N/A

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

   23. lower--.f64 98.4

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

4. Applied rewrites 98.4%

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

5. Final simplification 98.4%

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

Alternative 2: 65.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{a}\\ t_2 := \frac{z - t}{a - t} \cdot y\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+230}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* z y) a)) (t_2 (* (/ (- z t) (- a t)) y)))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 4e+230) (+ y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * y) / a;
	double t_2 = ((z - t) / (a - t)) * y;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 4e+230) {
		tmp = y + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * y) / a;
	double t_2 = ((z - t) / (a - t)) * y;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 4e+230) {
		tmp = y + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z * y) / a
	t_2 = ((z - t) / (a - t)) * y
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 4e+230:
		tmp = y + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * y) / a)
	t_2 = Float64(Float64(Float64(z - t) / Float64(a - t)) * y)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 4e+230)
		tmp = Float64(y + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * y) / a;
	t_2 = ((z - t) / (a - t)) * y;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 4e+230)
		tmp = y + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 4e+230], N[(y + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t))) < -inf.0 or 4.0000000000000004e230 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t)))

   1. Initial program 90.5%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 73.7%

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

   if -inf.0 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t))) < 4.0000000000000004e230

   1. Initial program 99.5%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 69.3

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

   5. Applied rewrites 69.3%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \cdot y \leq -\infty:\\ \;\;\;\;\frac{z \cdot y}{a}\\ \mathbf{elif}\;\frac{z - t}{a - t} \cdot y \leq 4 \cdot 10^{+230}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{a} \cdot y\\ t_2 := \frac{z - t}{a - t} \cdot y\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+230}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ z a) y)) (t_2 (* (/ (- z t) (- a t)) y)))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 4e+230) (+ y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z / a) * y;
	double t_2 = ((z - t) / (a - t)) * y;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 4e+230) {
		tmp = y + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z / a) * y;
	double t_2 = ((z - t) / (a - t)) * y;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 4e+230) {
		tmp = y + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z / a) * y
	t_2 = ((z - t) / (a - t)) * y
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 4e+230:
		tmp = y + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z / a) * y)
	t_2 = Float64(Float64(Float64(z - t) / Float64(a - t)) * y)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 4e+230)
		tmp = Float64(y + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z / a) * y;
	t_2 = ((z - t) / (a - t)) * y;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 4e+230)
		tmp = y + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / a), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 4e+230], N[(y + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t))) < -inf.0 or 4.0000000000000004e230 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t)))

   1. Initial program 90.5%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 73.7%

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

   10. Applied rewrites 70.5%

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

   if -inf.0 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t))) < 4.0000000000000004e230

   1. Initial program 99.5%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 69.3

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

   5. Applied rewrites 69.3%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \cdot y \leq -\infty:\\ \;\;\;\;\frac{z}{a} \cdot y\\ \mathbf{elif}\;\frac{z - t}{a - t} \cdot y \leq 4 \cdot 10^{+230}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 0.002)
     (fma (/ z a) y x)
     (if (<= t_1 1.0) (+ y x) (fma (/ (- z) t) y x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= 0.002) {
		tmp = fma((z / a), y, x);
	} else if (t_1 <= 1.0) {
		tmp = y + x;
	} else {
		tmp = fma((-z / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= 0.002)
		tmp = fma(Float64(z / a), y, x);
	elseif (t_1 <= 1.0)
		tmp = Float64(y + x);
	else
		tmp = fma(Float64(Float64(-z) / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.002], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(y + x), $MachinePrecision], N[(N[((-z) / t), $MachinePrecision] * y + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 2e-3

   1. Initial program 98.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 77.8

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

   5. Applied rewrites 77.8%

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

   if 2e-3 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 1 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 94.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 75.5%

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

Alternative 5: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+87}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 0.002)
     (fma (/ z a) y x)
     (if (<= t_1 5e+87) (+ y x) (* (/ y (- a t)) z)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= 0.002) {
		tmp = fma((z / a), y, x);
	} else if (t_1 <= 5e+87) {
		tmp = y + x;
	} else {
		tmp = (y / (a - t)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= 0.002)
		tmp = fma(Float64(z / a), y, x);
	elseif (t_1 <= 5e+87)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(y / Float64(a - t)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.002], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+87], N[(y + x), $MachinePrecision], N[(N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 2e-3

   1. Initial program 98.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 77.8

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

   5. Applied rewrites 77.8%

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

   if 2e-3 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999998e87

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 95.6

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

   5. Applied rewrites 95.6%

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

   if 4.9999999999999998e87 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 92.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 77.8

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

   5. Applied rewrites 77.8%

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

Alternative 6: 79.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+87}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 0.002)
     (fma (/ z a) y x)
     (if (<= t_1 5e+87) (+ y x) (* (/ y (- t)) z)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= 0.002) {
		tmp = fma((z / a), y, x);
	} else if (t_1 <= 5e+87) {
		tmp = y + x;
	} else {
		tmp = (y / -t) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= 0.002)
		tmp = fma(Float64(z / a), y, x);
	elseif (t_1 <= 5e+87)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(y / Float64(-t)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.002], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+87], N[(y + x), $MachinePrecision], N[(N[(y / (-t)), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 2e-3

   1. Initial program 98.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 77.8

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

   5. Applied rewrites 77.8%

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

   if 2e-3 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999998e87

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 95.6

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

   5. Applied rewrites 95.6%

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

   if 4.9999999999999998e87 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 92.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 77.8

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{y}{-1 \cdot t} \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 59.2%

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

Alternative 7: 79.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+87}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-y\right) \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 0.002)
     (fma (/ z a) y x)
     (if (<= t_1 5e+87) (+ y x) (/ (* (- y) z) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= 0.002) {
		tmp = fma((z / a), y, x);
	} else if (t_1 <= 5e+87) {
		tmp = y + x;
	} else {
		tmp = (-y * z) / t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= 0.002)
		tmp = fma(Float64(z / a), y, x);
	elseif (t_1 <= 5e+87)
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(Float64(-y) * z) / t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.002], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+87], N[(y + x), $MachinePrecision], N[(N[((-y) * z), $MachinePrecision] / t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 2e-3

   1. Initial program 98.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 77.8

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

   5. Applied rewrites 77.8%

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

   if 2e-3 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999998e87

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 95.6

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

   5. Applied rewrites 95.6%

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

   if 4.9999999999999998e87 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 92.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 59.2%

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

   9. Taylor expanded in t around 0

      \[\leadsto \frac{-1 \cdot \left(y \cdot z\right)}{t} \]
   10. Step-by-step derivation

   11. Applied rewrites 59.2%

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

Alternative 8: 80.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+107}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 0.002)
     (fma (/ z a) y x)
     (if (<= t_1 5e+107) (+ y x) (fma z (/ y a) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= 0.002) {
		tmp = fma((z / a), y, x);
	} else if (t_1 <= 5e+107) {
		tmp = y + x;
	} else {
		tmp = fma(z, (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= 0.002)
		tmp = fma(Float64(z / a), y, x);
	elseif (t_1 <= 5e+107)
		tmp = Float64(y + x);
	else
		tmp = fma(z, Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.002], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+107], N[(y + x), $MachinePrecision], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 2e-3

   1. Initial program 98.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 77.8

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

   5. Applied rewrites 77.8%

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

   if 2e-3 < (/.f64 (-.f64 z t) (-.f64 a t)) < 5.0000000000000002e107

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 92.7

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

   5. Applied rewrites 92.7%

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

   if 5.0000000000000002e107 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. associate--r+ N/A

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

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 90.6

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

   4. Applied rewrites 90.6%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 60.9

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

   7. Applied rewrites 60.9%

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

Alternative 9: 80.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{if}\;t\_1 \leq 0.002:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+107}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (fma (/ z a) y x)))
   (if (<= t_1 0.002) t_2 (if (<= t_1 5e+107) (+ y x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = fma((z / a), y, x);
	double tmp;
	if (t_1 <= 0.002) {
		tmp = t_2;
	} else if (t_1 <= 5e+107) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = fma(Float64(z / a), y, x)
	tmp = 0.0
	if (t_1 <= 0.002)
		tmp = t_2;
	elseif (t_1 <= 5e+107)
		tmp = Float64(y + x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, 0.002], t$95$2, If[LessEqual[t$95$1, 5e+107], N[(y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 2e-3 or 5.0000000000000002e107 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 97.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 75.1

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

   5. Applied rewrites 75.1%

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

   if 2e-3 < (/.f64 (-.f64 z t) (-.f64 a t)) < 5.0000000000000002e107

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 92.7

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

   5. Applied rewrites 92.7%

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

Alternative 10: 86.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- z t) (- a t)) 0.002)
   (fma (/ (- z t) a) y x)
   (fma (/ (- t z) t) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (a - t)) <= 0.002) {
		tmp = fma(((z - t) / a), y, x);
	} else {
		tmp = fma(((t - z) / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z - t) / Float64(a - t)) <= 0.002)
		tmp = fma(Float64(Float64(z - t) / a), y, x);
	else
		tmp = fma(Float64(Float64(t - z) / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], 0.002], N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(N[(t - z), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 2e-3

   1. Initial program 98.6%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 89.4

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

   5. Applied rewrites 89.4%

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

   if 2e-3 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 98.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 92.2%

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

Alternative 11: 81.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- z t) (- a t)) 0.002) (fma (/ z a) y x) (fma (/ (- t z) t) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (a - t)) <= 0.002) {
		tmp = fma((z / a), y, x);
	} else {
		tmp = fma(((t - z) / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z - t) / Float64(a - t)) <= 0.002)
		tmp = fma(Float64(z / a), y, x);
	else
		tmp = fma(Float64(Float64(t - z) / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], 0.002], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(N[(t - z), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 2e-3

   1. Initial program 98.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 77.8

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

   5. Applied rewrites 77.8%

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

   if 2e-3 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 98.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 92.2%

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

Alternative 12: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

3. Final simplification 98.4%

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

Alternative 13: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 98.4

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

   6. lift-/.f64 N/A

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

   7. frac-2neg N/A

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

   8. lower-/.f64 N/A

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

   9. neg-sub0 N/A

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

   10. lift--.f64 N/A

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

   11. sub-neg N/A

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

   12. +-commutative N/A

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

   13. associate--r+ N/A

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

   14. neg-sub0 N/A

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

   15. remove-double-neg N/A

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

   16. lower--.f64 N/A

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

   17. neg-sub0 N/A

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

   18. lift--.f64 N/A

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

   19. sub-neg N/A

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

   20. +-commutative N/A

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

   21. associate--r+ N/A

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

   22. neg-sub0 N/A

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

   23. remove-double-neg N/A

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

   24. lower--.f64 98.4

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

4. Applied rewrites 98.4%

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

Alternative 14: 60.9% 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 98.4%

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

3. Taylor expanded in t around inf

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

   1. +-commutative N/A

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

   2. lower-+.f64 61.8

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

5. Applied rewrites 61.8%

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

Developer Target 1: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + y \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* y (/ (- z t) (- a t))))))
   (if (< y -8.508084860551241e-17)
     t_1
     (if (< y 2.894426862792089e-49)
       (+ x (* (* y (- z t)) (/ 1.0 (- a t))))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double tmp;
	if (y < -8.508084860551241e-17) {
		tmp = t_1;
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	} 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 - t) / (a - t)))
    if (y < (-8.508084860551241d-17)) then
        tmp = t_1
    else if (y < 2.894426862792089d-49) then
        tmp = x + ((y * (z - t)) * (1.0d0 / (a - t)))
    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 - t) / (a - t)));
	double tmp;
	if (y < -8.508084860551241e-17) {
		tmp = t_1;
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (y * ((z - t) / (a - t)))
	tmp = 0
	if y < -8.508084860551241e-17:
		tmp = t_1
	elif y < 2.894426862792089e-49:
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(y * Float64(Float64(z - t) / Float64(a - t))))
	tmp = 0.0
	if (y < -8.508084860551241e-17)
		tmp = t_1;
	elseif (y < 2.894426862792089e-49)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) * Float64(1.0 / Float64(a - t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (y * ((z - t) / (a - t)));
	tmp = 0.0;
	if (y < -8.508084860551241e-17)
		tmp = t_1;
	elseif (y < 2.894426862792089e-49)
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -8.508084860551241e-17], t$95$1, If[Less[y, 2.894426862792089e-49], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024255 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -8508084860551241/100000000000000000000000000000000) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2894426862792089/10000000000000000000000000000000000000000000000000000000000000000) (+ x (* (* y (- z t)) (/ 1 (- a t)))) (+ x (* y (/ (- z t) (- a t)))))))

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