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

Percentage Accurate: 97.6% → 97.7%

Time: 10.4s

Alternatives: 10

Speedup: 1.1×

• 25.6% 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: 97.6% 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: 97.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

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) / (t / z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

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

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

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

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

   5. /-lowering-/.f64 99.5

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

4. Applied egg-rr 99.5%

   \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}} \]
5. Add Preprocessing

Alternative 2: 94.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y x) (/ z t))))
   (if (<= (/ z t) -2e+30) t_1 (if (<= (/ z t) 2e-29) (fma (/ z t) y x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - x) * (z / t);
	double tmp;
	if ((z / t) <= -2e+30) {
		tmp = t_1;
	} else if ((z / t) <= 2e-29) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - x) * Float64(z / t))
	tmp = 0.0
	if (Float64(z / t) <= -2e+30)
		tmp = t_1;
	elseif (Float64(z / t) <= 2e-29)
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z / t), $MachinePrecision], -2e+30], t$95$1, If[LessEqual[N[(z / t), $MachinePrecision], 2e-29], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z t) < -2e30 or 1.99999999999999989e-29 < (/.f64 z t)

   1. Initial program 99.7%

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

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

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

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

      4. --lowering--.f64 92.2

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

   5. Simplified 92.2%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

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

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

      7. --lowering--.f64 99.0

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

   7. Applied egg-rr 99.0%

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

   if -2e30 < (/.f64 z t) < 1.99999999999999989e-29

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 99.2%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 99.2

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

   7. Applied egg-rr 99.2%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{+30}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{y - x}{t}\\ \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (/ (- y x) t))))
   (if (<= (/ z t) -2e-12) t_1 (if (<= (/ z t) 2e-29) (fma (/ z t) y x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * ((y - x) / t);
	double tmp;
	if ((z / t) <= -2e-12) {
		tmp = t_1;
	} else if ((z / t) <= 2e-29) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(Float64(y - x) / t))
	tmp = 0.0
	if (Float64(z / t) <= -2e-12)
		tmp = t_1;
	elseif (Float64(z / t) <= 2e-29)
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z / t), $MachinePrecision], -2e-12], t$95$1, If[LessEqual[N[(z / t), $MachinePrecision], 2e-29], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z t) < -1.99999999999999996e-12 or 1.99999999999999989e-29 < (/.f64 z t)

   1. Initial program 99.7%

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

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

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

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

      4. --lowering--.f64 92.4

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

   5. Simplified 92.4%

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

   if -1.99999999999999996e-12 < (/.f64 z t) < 1.99999999999999989e-29

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 99.2%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 99.2

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

   7. Applied egg-rr 99.2%

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

Alternative 4: 64.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z}{t}\\ \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-97}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (/ z t))))
   (if (<= (/ z t) -1e-22) t_1 (if (<= (/ z t) 5e-97) x t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (z / t);
	double tmp;
	if ((z / t) <= -1e-22) {
		tmp = t_1;
	} else if ((z / t) <= 5e-97) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	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) :: tmp
    t_1 = y * (z / t)
    if ((z / t) <= (-1d-22)) then
        tmp = t_1
    else if ((z / t) <= 5d-97) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y * (z / t);
	double tmp;
	if ((z / t) <= -1e-22) {
		tmp = t_1;
	} else if ((z / t) <= 5e-97) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * (z / t)
	tmp = 0
	if (z / t) <= -1e-22:
		tmp = t_1
	elif (z / t) <= 5e-97:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(z / t))
	tmp = 0.0
	if (Float64(z / t) <= -1e-22)
		tmp = t_1;
	elseif (Float64(z / t) <= 5e-97)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * (z / t);
	tmp = 0.0;
	if ((z / t) <= -1e-22)
		tmp = t_1;
	elseif ((z / t) <= 5e-97)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z / t), $MachinePrecision], -1e-22], t$95$1, If[LessEqual[N[(z / t), $MachinePrecision], 5e-97], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z t) < -1e-22 or 4.9999999999999995e-97 < (/.f64 z t)

   1. Initial program 99.7%

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

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

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

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

      4. --lowering--.f64 88.4

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

   5. Simplified 88.4%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 55.2%

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

      1. clear-num N/A

         \[\leadsto z \cdot \color{blue}{\frac{1}{\frac{t}{y}}} \]

      2. associate-/r/ N/A

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

      3. associate-*r* N/A

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

      4. div-inv N/A

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

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

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

      6. /-lowering-/.f64 61.8

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

   10. Applied egg-rr 61.8%

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

   if -1e-22 < (/.f64 z t) < 4.9999999999999995e-97

   1. Initial program 99.1%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 86.9%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{-22}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-97}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{y}{t}\\ \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-97}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (/ y t))))
   (if (<= (/ z t) -1e-22) t_1 (if (<= (/ z t) 5e-97) x t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (y / t);
	double tmp;
	if ((z / t) <= -1e-22) {
		tmp = t_1;
	} else if ((z / t) <= 5e-97) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	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) :: tmp
    t_1 = z * (y / t)
    if ((z / t) <= (-1d-22)) then
        tmp = t_1
    else if ((z / t) <= 5d-97) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = z * (y / t);
	double tmp;
	if ((z / t) <= -1e-22) {
		tmp = t_1;
	} else if ((z / t) <= 5e-97) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * (y / t)
	tmp = 0
	if (z / t) <= -1e-22:
		tmp = t_1
	elif (z / t) <= 5e-97:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(y / t))
	tmp = 0.0
	if (Float64(z / t) <= -1e-22)
		tmp = t_1;
	elseif (Float64(z / t) <= 5e-97)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = z * (y / t);
	tmp = 0.0;
	if ((z / t) <= -1e-22)
		tmp = t_1;
	elseif ((z / t) <= 5e-97)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z / t), $MachinePrecision], -1e-22], t$95$1, If[LessEqual[N[(z / t), $MachinePrecision], 5e-97], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z t) < -1e-22 or 4.9999999999999995e-97 < (/.f64 z t)

   1. Initial program 99.7%

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

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

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

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

      4. --lowering--.f64 88.4

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

   5. Simplified 88.4%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 55.2%

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

   if -1e-22 < (/.f64 z t) < 4.9999999999999995e-97

   1. Initial program 99.1%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 86.9%

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

Alternative 6: 76.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq 10000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{-t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ z t) 10000000.0) (fma (/ z t) y x) (* x (/ z (- t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z / t) <= 10000000.0) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = x * (z / -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z / t) <= 10000000.0)
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = Float64(x * Float64(z / Float64(-t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z / t), $MachinePrecision], 10000000.0], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], N[(x * N[(z / (-t)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z t) < 1e7

   1. Initial program 99.3%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 89.7%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 89.7

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

   7. Applied egg-rr 89.7%

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

   if 1e7 < (/.f64 z t)

   1. Initial program 99.7%

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

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

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

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

      4. --lowering--.f64 90.5

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

   5. Simplified 90.5%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 61.2

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

   8. Simplified 61.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-*r* N/A

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

      4. div-inv N/A

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

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

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

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

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

      7. neg-lowering-neg.f64 65.8

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

   10. Applied egg-rr 65.8%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq 10000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{-t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;-z \cdot \frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ z t) 1e+59) (fma (/ z t) y x) (- (* z (/ x t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z / t) <= 1e+59) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = -(z * (x / t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z / t) <= 1e+59)
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = Float64(-Float64(z * Float64(x / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z / t), $MachinePrecision], 1e+59], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], (-N[(z * N[(x / t), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z t) < 9.99999999999999972e58

   1. Initial program 99.3%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 88.2%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 88.2

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

   7. Applied egg-rr 88.2%

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

   if 9.99999999999999972e58 < (/.f64 z t)

   1. Initial program 99.8%

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

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

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

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

      4. --lowering--.f64 96.3

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

   5. Simplified 96.3%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 68.1

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

   8. Simplified 68.1%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;-z \cdot \frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. --lowering--.f64 99.4

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

4. Applied egg-rr 99.4%

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

Alternative 9: 77.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in y around inf

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

5. Simplified 81.5%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

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

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

   4. /-lowering-/.f64 81.5

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

7. Applied egg-rr 81.5%

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

Alternative 10: 38.4% accurate, 23.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return 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 = x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := x

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in z around 0

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

5. Simplified 43.3%

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

Developer Target 1: 97.4% 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 2024205 
(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))))