Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4

Percentage Accurate: 97.7% → 98.0%

Time: 5.8s

Alternatives: 11

Speedup: 0.6×

• 25.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ z t) 1e+96) (fma (/ z t) (- y x) x) (/ (* (- y x) z) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z / t) <= 1e+96) {
		tmp = fma((z / t), (y - x), x);
	} else {
		tmp = ((y - x) * z) / t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z / t) <= 1e+96)
		tmp = fma(Float64(z / t), Float64(y - x), x);
	else
		tmp = Float64(Float64(Float64(y - x) * z) / t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z / t), $MachinePrecision], 1e+96], N[(N[(z / t), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z t) < 1.00000000000000005e96

   1. Initial program 98.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.6

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

   4. Applied rewrites 98.6%

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

   if 1.00000000000000005e96 < (/.f64 z t)

   1. Initial program 86.9%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 97.2

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

   5. Applied rewrites 97.2%

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

   7. Applied rewrites 100.0%

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

Alternative 2: 92.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2000000 \lor \neg \left(\frac{z}{t} \leq 2 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ z t) -2000000.0) (not (<= (/ z t) 2e+44)))
   (* (/ (- y x) t) z)
   (+ x (/ (* y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z / t) <= -2000000.0) || !((z / t) <= 2e+44)) {
		tmp = ((y - x) / t) * z;
	} else {
		tmp = x + ((y * z) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z / t) <= (-2000000.0d0)) .or. (.not. ((z / t) <= 2d+44))) then
        tmp = ((y - x) / t) * z
    else
        tmp = x + ((y * z) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z / t) <= -2000000.0) || !((z / t) <= 2e+44)) {
		tmp = ((y - x) / t) * z;
	} else {
		tmp = x + ((y * z) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z / t) <= -2000000.0) or not ((z / t) <= 2e+44):
		tmp = ((y - x) / t) * z
	else:
		tmp = x + ((y * z) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z / t) <= -2000000.0) || !(Float64(z / t) <= 2e+44))
		tmp = Float64(Float64(Float64(y - x) / t) * z);
	else
		tmp = Float64(x + Float64(Float64(y * z) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z / t) <= -2000000.0) || ~(((z / t) <= 2e+44)))
		tmp = ((y - x) / t) * z;
	else
		tmp = x + ((y * z) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z / t), $MachinePrecision], -2000000.0], N[Not[LessEqual[N[(z / t), $MachinePrecision], 2e+44]], $MachinePrecision]], N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision], N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z t) < -2e6 or 2.0000000000000002e44 < (/.f64 z t)

   1. Initial program 94.8%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 96.5

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

   5. Applied rewrites 96.5%

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

   if -2e6 < (/.f64 z t) < 2.0000000000000002e44

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 93.9

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

   4. Applied rewrites 93.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 92.0

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

   7. Applied rewrites 92.0%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2000000 \lor \neg \left(\frac{z}{t} \leq 2 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2000000 \lor \neg \left(\frac{z}{t} \leq 10^{-8}\right):\\ \;\;\;\;\frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ z t) -2000000.0) (not (<= (/ z t) 1e-8)))
   (* (/ (- y x) t) z)
   (fma z (/ y t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z / t) <= -2000000.0) || !((z / t) <= 1e-8)) {
		tmp = ((y - x) / t) * z;
	} else {
		tmp = fma(z, (y / t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z / t) <= -2000000.0) || !(Float64(z / t) <= 1e-8))
		tmp = Float64(Float64(Float64(y - x) / t) * z);
	else
		tmp = fma(z, Float64(y / t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z / t), $MachinePrecision], -2000000.0], N[Not[LessEqual[N[(z / t), $MachinePrecision], 1e-8]], $MachinePrecision]], N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision], N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z t) < -2e6 or 1e-8 < (/.f64 z t)

   1. Initial program 95.0%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if -2e6 < (/.f64 z t) < 1e-8

   1. Initial program 99.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 91.9

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

   4. Applied rewrites 91.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 93.1

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

   7. Applied rewrites 93.1%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2000000 \lor \neg \left(\frac{z}{t} \leq 10^{-8}\right):\\ \;\;\;\;\frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{-96} \lor \neg \left(\frac{z}{t} \leq 5 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ z t) -5e-96) (not (<= (/ z t) 5e-19)))
   (* (/ y t) z)
   (* 1.0 x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z / t) <= -5e-96) || !((z / t) <= 5e-19)) {
		tmp = (y / t) * z;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z / t) <= (-5d-96)) .or. (.not. ((z / t) <= 5d-19))) then
        tmp = (y / t) * z
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z / t) <= -5e-96) || !((z / t) <= 5e-19)) {
		tmp = (y / t) * z;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z / t) <= -5e-96) or not ((z / t) <= 5e-19):
		tmp = (y / t) * z
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z / t) <= -5e-96) || !(Float64(z / t) <= 5e-19))
		tmp = Float64(Float64(y / t) * z);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z / t) <= -5e-96) || ~(((z / t) <= 5e-19)))
		tmp = (y / t) * z;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z / t), $MachinePrecision], -5e-96], N[Not[LessEqual[N[(z / t), $MachinePrecision], 5e-19]], $MachinePrecision]], N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z t) < -4.99999999999999995e-96 or 5.0000000000000004e-19 < (/.f64 z t)

   1. Initial program 95.5%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 55.4

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

   5. Applied rewrites 55.4%

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

   if -4.99999999999999995e-96 < (/.f64 z t) < 5.0000000000000004e-19

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 82.1

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in z around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 82.1%

      \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{-96} \lor \neg \left(\frac{z}{t} \leq 5 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{-96}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-19}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ z t) -5e-96)
   (* (/ z t) y)
   (if (<= (/ z t) 5e-19) (* 1.0 x) (* (/ y t) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z / t) <= -5e-96) {
		tmp = (z / t) * y;
	} else if ((z / t) <= 5e-19) {
		tmp = 1.0 * x;
	} else {
		tmp = (y / t) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z / t) <= (-5d-96)) then
        tmp = (z / t) * y
    else if ((z / t) <= 5d-19) then
        tmp = 1.0d0 * x
    else
        tmp = (y / t) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z / t) <= -5e-96) {
		tmp = (z / t) * y;
	} else if ((z / t) <= 5e-19) {
		tmp = 1.0 * x;
	} else {
		tmp = (y / t) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z / t) <= -5e-96:
		tmp = (z / t) * y
	elif (z / t) <= 5e-19:
		tmp = 1.0 * x
	else:
		tmp = (y / t) * z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z / t) <= -5e-96)
		tmp = Float64(Float64(z / t) * y);
	elseif (Float64(z / t) <= 5e-19)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(y / t) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z / t) <= -5e-96)
		tmp = (z / t) * y;
	elseif ((z / t) <= 5e-19)
		tmp = 1.0 * x;
	else
		tmp = (y / t) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z / t), $MachinePrecision], -5e-96], N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[N[(z / t), $MachinePrecision], 5e-19], N[(1.0 * x), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 z t) < -4.99999999999999995e-96

   1. Initial program 97.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 58.3

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 62.2%

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

   if -4.99999999999999995e-96 < (/.f64 z t) < 5.0000000000000004e-19

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 82.1

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in z around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 82.1%

      \[\leadsto 1 \cdot x \]

   if 5.0000000000000004e-19 < (/.f64 z t)

   1. Initial program 92.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 51.1

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

   5. Applied rewrites 51.1%

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

Alternative 6: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq 2 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ z t) 2e+44) (fma z (/ y t) x) (* (/ (- x) t) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z / t) <= 2e+44) {
		tmp = fma(z, (y / t), x);
	} else {
		tmp = (-x / t) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z / t) <= 2e+44)
		tmp = fma(z, Float64(y / t), x);
	else
		tmp = Float64(Float64(Float64(-x) / t) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z / t), $MachinePrecision], 2e+44], N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision], N[(N[((-x) / t), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z t) < 2.0000000000000002e44

   1. Initial program 98.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 92.5

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

   4. Applied rewrites 92.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 79.2

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

   7. Applied rewrites 79.2%

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

   if 2.0000000000000002e44 < (/.f64 z t)

   1. Initial program 90.8%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 59.9%

      \[\leadsto \frac{-x}{t} \cdot z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 95.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-175} \lor \neg \left(z \leq 6.5 \cdot 10^{-143}\right):\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -8.5e-175) (not (<= z 6.5e-143)))
   (fma z (/ (- y x) t) x)
   (+ x (/ (* y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -8.5e-175) || !(z <= 6.5e-143)) {
		tmp = fma(z, ((y - x) / t), x);
	} else {
		tmp = x + ((y * z) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -8.5e-175) || !(z <= 6.5e-143))
		tmp = fma(z, Float64(Float64(y - x) / t), x);
	else
		tmp = Float64(x + Float64(Float64(y * z) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -8.5e-175], N[Not[LessEqual[z, 6.5e-143]], $MachinePrecision]], N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.5000000000000005e-175 or 6.4999999999999999e-143 < z

   1. Initial program 96.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 98.0

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

   4. Applied rewrites 98.0%

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

   if -8.5000000000000005e-175 < z < 6.4999999999999999e-143

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 96.0

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

   7. Applied rewrites 96.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-175} \lor \neg \left(z \leq 6.5 \cdot 10^{-143}\right):\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+44} \lor \neg \left(x \leq 2.55 \cdot 10^{-12}\right):\\ \;\;\;\;\left(1 - \frac{z}{t}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.4e+44) (not (<= x 2.55e-12)))
   (* (- 1.0 (/ z t)) x)
   (fma z (/ y t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.4e+44) || !(x <= 2.55e-12)) {
		tmp = (1.0 - (z / t)) * x;
	} else {
		tmp = fma(z, (y / t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.4e+44) || !(x <= 2.55e-12))
		tmp = Float64(Float64(1.0 - Float64(z / t)) * x);
	else
		tmp = fma(z, Float64(y / t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.4e+44], N[Not[LessEqual[x, 2.55e-12]], $MachinePrecision]], N[(N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.39999999999999991e44 or 2.54999999999999984e-12 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 88.4

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

   5. Applied rewrites 88.4%

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

   if -4.39999999999999991e44 < x < 2.54999999999999984e-12

   1. Initial program 94.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 96.4

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

   4. Applied rewrites 96.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 82.3

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

   7. Applied rewrites 82.3%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+44} \lor \neg \left(x \leq 2.55 \cdot 10^{-12}\right):\\ \;\;\;\;\left(1 - \frac{z}{t}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.6 \cdot 10^{+208}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{t} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 7.6e+208) (fma z (/ y t) x) (* (/ (- z) t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 7.6e+208) {
		tmp = fma(z, (y / t), x);
	} else {
		tmp = (-z / t) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 7.6e+208)
		tmp = fma(z, Float64(y / t), x);
	else
		tmp = Float64(Float64(Float64(-z) / t) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 7.6e+208], N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision], N[(N[((-z) / t), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.6000000000000004e208

   1. Initial program 96.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 93.5

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

   4. Applied rewrites 93.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 74.9

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

   7. Applied rewrites 74.9%

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

   if 7.6000000000000004e208 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 94.8

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

   5. Applied rewrites 94.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 73.0%

      \[\leadsto \frac{-z}{t} \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 73.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 93.6

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

4. Applied rewrites 93.6%

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

5. Taylor expanded in x around 0

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

   1. lower-/.f64 72.8

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

7. Applied rewrites 72.8%

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

Alternative 11: 38.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* 1.0 x))

C

double code(double x, double y, double z, double t) {
	return 1.0 * x;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 * x
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0 * x;
}

Python

def code(x, y, z, t):
	return 1.0 * x

Julia

function code(x, y, z, t)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_, z_, t_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 96.9%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. *-lft-identity N/A

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

   6. lower--.f64 N/A

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

   7. lower-/.f64 63.0

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

5. Applied rewrites 63.0%

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

6. Taylor expanded in z around 0

   \[\leadsto 1 \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 35.6%

   \[\leadsto 1 \cdot x \]
9. Add Preprocessing

Developer Target 1: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - x\right) \cdot \frac{z}{t}\\ t_2 := x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{if}\;t\_1 < -1013646692435.8867:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 < 0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y x) (/ z t))) (t_2 (+ x (/ (- y x) (/ t z)))))
   (if (< t_1 -1013646692435.8867)
     t_2
     (if (< t_1 0.0) (+ x (/ (* (- y x) z) t)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - x) * (z / t);
	double t_2 = x + ((y - x) / (t / z));
	double tmp;
	if (t_1 < -1013646692435.8867) {
		tmp = t_2;
	} else if (t_1 < 0.0) {
		tmp = x + (((y - x) * z) / t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (y - x) * (z / t)
    t_2 = x + ((y - x) / (t / z))
    if (t_1 < (-1013646692435.8867d0)) then
        tmp = t_2
    else if (t_1 < 0.0d0) then
        tmp = x + (((y - x) * z) / t)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y - x) * (z / t);
	double t_2 = x + ((y - x) / (t / z));
	double tmp;
	if (t_1 < -1013646692435.8867) {
		tmp = t_2;
	} else if (t_1 < 0.0) {
		tmp = x + (((y - x) * z) / t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y - x) * (z / t)
	t_2 = x + ((y - x) / (t / z))
	tmp = 0
	if t_1 < -1013646692435.8867:
		tmp = t_2
	elif t_1 < 0.0:
		tmp = x + (((y - x) * z) / t)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - x) * Float64(z / t))
	t_2 = Float64(x + Float64(Float64(y - x) / Float64(t / z)))
	tmp = 0.0
	if (t_1 < -1013646692435.8867)
		tmp = t_2;
	elseif (t_1 < 0.0)
		tmp = Float64(x + Float64(Float64(Float64(y - x) * z) / t));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y - x) * (z / t);
	t_2 = x + ((y - x) / (t / z));
	tmp = 0.0;
	if (t_1 < -1013646692435.8867)
		tmp = t_2;
	elseif (t_1 < 0.0)
		tmp = x + (((y - x) * z) / t);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - x), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$1, -1013646692435.8867], t$95$2, If[Less[t$95$1, 0.0], N[(x + N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024339 
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* (- y x) (/ z t)) -10136466924358867/10000) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) 0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z))))))

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