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

Percentage Accurate: 97.7% → 97.7%

Time: 8.9s

Alternatives: 8

Speedup: 1.0×

• 25.2% 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: 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]

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 93.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -20 \lor \neg \left(\frac{z}{t} \leq 2 \cdot 10^{+17}\right):\\ \;\;\;\;z \cdot \left(\frac{y}{t} - \frac{x}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ z t) -20.0) (not (<= (/ z t) 2e+17)))
   (* z (- (/ y t) (/ x t)))
   (+ x (* y (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z / t) <= -20.0) || !((z / t) <= 2e+17)) {
		tmp = z * ((y / t) - (x / t));
	} 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) <= (-20.0d0)) .or. (.not. ((z / t) <= 2d+17))) then
        tmp = z * ((y / t) - (x / t))
    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) <= -20.0) || !((z / t) <= 2e+17)) {
		tmp = z * ((y / t) - (x / t));
	} else {
		tmp = x + (y * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z / t) <= -20.0) or not ((z / t) <= 2e+17):
		tmp = z * ((y / t) - (x / t))
	else:
		tmp = x + (y * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z / t) <= -20.0) || !(Float64(z / t) <= 2e+17))
		tmp = Float64(z * Float64(Float64(y / t) - Float64(x / t)));
	else
		tmp = Float64(x + Float64(y * Float64(z / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z / t) <= -20.0) || ~(((z / t) <= 2e+17)))
		tmp = z * ((y / t) - (x / t));
	else
		tmp = x + (y * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z / t), $MachinePrecision], -20.0], N[Not[LessEqual[N[(z / t), $MachinePrecision], 2e+17]], $MachinePrecision]], N[(z * N[(N[(y / t), $MachinePrecision] - N[(x / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z t) < -20 or 2e17 < (/.f64 z t)

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 94.7%

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

   if -20 < (/.f64 z t) < 2e17

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 91.8%

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

      1. associate-*r/ 98.0%

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

   4. Simplified 98.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -20 \lor \neg \left(\frac{z}{t} \leq 2 \cdot 10^{+17}\right):\\ \;\;\;\;z \cdot \left(\frac{y}{t} - \frac{x}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 3: 64.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -20 \lor \neg \left(\frac{z}{t} \leq 2 \cdot 10^{+17}\right):\\ \;\;\;\;x \cdot \frac{-z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ z t) -20.0) (not (<= (/ z t) 2e+17))) (* x (/ (- z) t)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z / t) <= -20.0) || !((z / t) <= 2e+17)) {
		tmp = x * (-z / t);
	} else {
		tmp = 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) <= (-20.0d0)) .or. (.not. ((z / t) <= 2d+17))) then
        tmp = x * (-z / t)
    else
        tmp = 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) <= -20.0) || !((z / t) <= 2e+17)) {
		tmp = x * (-z / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z / t) <= -20.0) or not ((z / t) <= 2e+17):
		tmp = x * (-z / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z / t) <= -20.0) || !(Float64(z / t) <= 2e+17))
		tmp = Float64(x * Float64(Float64(-z) / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z / t) <= -20.0) || ~(((z / t) <= 2e+17)))
		tmp = x * (-z / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z / t), $MachinePrecision], -20.0], N[Not[LessEqual[N[(z / t), $MachinePrecision], 2e+17]], $MachinePrecision]], N[(x * N[((-z) / t), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z t) < -20 or 2e17 < (/.f64 z t)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf 60.0%

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

      1. *-commutative 60.0%

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

      2. mul-1-neg 60.0%

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

      3. unsub-neg 60.0%

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

      4. distribute-lft-out-- 60.0%

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

      5. *-rgt-identity 60.0%

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

   4. Simplified 60.0%

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

      1. clear-num 60.0%

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

      2. div-inv 59.2%

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

   6. Applied egg-rr 59.2%

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

   7. Taylor expanded in t around 0 52.7%

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

      1. mul-1-neg 52.7%

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

      2. associate-*r/ 53.5%

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

      3. distribute-rgt-neg-in 53.5%

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

      4. distribute-neg-frac 53.5%

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

   9. Simplified 53.5%

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

   10. Taylor expanded in z around 0 52.7%

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

       1. mul-1-neg 52.7%

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

       2. associate-*l/ 58.9%

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

       3. *-commutative 58.9%

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

       4. distribute-rgt-neg-in 58.9%

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

       5. distribute-neg-frac 58.9%

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

   12. Simplified 58.9%

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

   if -20 < (/.f64 z t) < 2e17

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 70.3%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -20 \lor \neg \left(\frac{z}{t} \leq 2 \cdot 10^{+17}\right):\\ \;\;\;\;x \cdot \frac{-z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+18} \lor \neg \left(x \leq 0.95\right):\\ \;\;\;\;x - x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1e+18) (not (<= x 0.95)))
   (- x (* x (/ z t)))
   (+ x (* y (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1e+18) || !(x <= 0.95)) {
		tmp = x - (x * (z / t));
	} 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 ((x <= (-1d+18)) .or. (.not. (x <= 0.95d0))) then
        tmp = x - (x * (z / t))
    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 ((x <= -1e+18) || !(x <= 0.95)) {
		tmp = x - (x * (z / t));
	} else {
		tmp = x + (y * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1e+18) or not (x <= 0.95):
		tmp = x - (x * (z / t))
	else:
		tmp = x + (y * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1e+18) || !(x <= 0.95))
		tmp = Float64(x - Float64(x * Float64(z / t)));
	else
		tmp = Float64(x + Float64(y * Float64(z / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1e+18) || ~((x <= 0.95)))
		tmp = x - (x * (z / t));
	else
		tmp = x + (y * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1e+18], N[Not[LessEqual[x, 0.95]], $MachinePrecision]], N[(x - N[(x * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e18 or 0.94999999999999996 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 92.2%

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

      1. *-commutative 92.2%

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

      2. mul-1-neg 92.2%

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

      3. unsub-neg 92.2%

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

      4. distribute-lft-out-- 92.2%

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

      5. *-rgt-identity 92.2%

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

   4. Simplified 92.2%

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

   if -1e18 < x < 0.94999999999999996

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 83.1%

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

      1. associate-*r/ 89.2%

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

   4. Simplified 89.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+18} \lor \neg \left(x \leq 0.95\right):\\ \;\;\;\;x - x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 5: 85.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -430000000000:\\ \;\;\;\;x - x \cdot \frac{z}{t}\\ \mathbf{elif}\;x \leq 19:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -430000000000.0)
   (- x (* x (/ z t)))
   (if (<= x 19.0) (+ x (* y (/ z t))) (- x (/ x (/ t z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -430000000000.0) {
		tmp = x - (x * (z / t));
	} else if (x <= 19.0) {
		tmp = x + (y * (z / t));
	} else {
		tmp = x - (x / (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 (x <= (-430000000000.0d0)) then
        tmp = x - (x * (z / t))
    else if (x <= 19.0d0) then
        tmp = x + (y * (z / t))
    else
        tmp = x - (x / (t / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -430000000000.0) {
		tmp = x - (x * (z / t));
	} else if (x <= 19.0) {
		tmp = x + (y * (z / t));
	} else {
		tmp = x - (x / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -430000000000.0:
		tmp = x - (x * (z / t))
	elif x <= 19.0:
		tmp = x + (y * (z / t))
	else:
		tmp = x - (x / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -430000000000.0)
		tmp = Float64(x - Float64(x * Float64(z / t)));
	elseif (x <= 19.0)
		tmp = Float64(x + Float64(y * Float64(z / t)));
	else
		tmp = Float64(x - Float64(x / Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -430000000000.0)
		tmp = x - (x * (z / t));
	elseif (x <= 19.0)
		tmp = x + (y * (z / t));
	else
		tmp = x - (x / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -430000000000.0], N[(x - N[(x * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 19.0], N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(x / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.3e11

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 92.2%

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

      1. *-commutative 92.2%

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

      2. mul-1-neg 92.2%

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

      3. unsub-neg 92.2%

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

      4. distribute-lft-out-- 92.3%

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

      5. *-rgt-identity 92.3%

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

   4. Simplified 92.3%

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

   if -4.3e11 < x < 19

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 83.1%

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

      1. associate-*r/ 89.2%

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

   4. Simplified 89.2%

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

   if 19 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 92.1%

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

      1. *-commutative 92.1%

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

      2. mul-1-neg 92.1%

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

      3. unsub-neg 92.1%

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

      4. distribute-lft-out-- 92.1%

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

      5. *-rgt-identity 92.1%

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

   4. Simplified 92.1%

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

      1. clear-num 92.1%

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

      2. div-inv 92.1%

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

   6. Applied egg-rr 92.1%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -430000000000:\\ \;\;\;\;x - x \cdot \frac{z}{t}\\ \mathbf{elif}\;x \leq 19:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{t}{z}}\\ \end{array} \]

Alternative 6: 39.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -\infty:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z / t) <= -Double.POSITIVE_INFINITY) {
		tmp = x * (z / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z / t) <= -math.inf:
		tmp = x * (z / t)
	else:
		tmp = x
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z / t) <= -Inf)
		tmp = x * (z / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 z t) < -inf.0

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 76.2%

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

      1. *-commutative 76.2%

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

      2. mul-1-neg 76.2%

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

      3. unsub-neg 76.2%

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

      4. distribute-lft-out-- 76.2%

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

      5. *-rgt-identity 76.2%

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

   4. Simplified 76.2%

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

      1. clear-num 76.2%

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

      2. div-inv 71.7%

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

   6. Applied egg-rr 71.7%

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

   7. Taylor expanded in t around 0 58.3%

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

      1. mul-1-neg 58.3%

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

      2. associate-*r/ 58.3%

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

      3. distribute-rgt-neg-in 58.3%

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

      4. distribute-neg-frac 58.3%

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

   9. Simplified 58.3%

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

       1. expm1-log1p-u 20.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(z \cdot \frac{-x}{t}\right)\right)} \]

       2. expm1-udef 20.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(z \cdot \frac{-x}{t}\right)} - 1} \]

       3. log1p-udef 20.0%

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

       4. add-exp-log 58.3%

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

       5. add-sqr-sqrt 38.3%

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

       6. sqrt-unprod 39.0%

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

       7. sqr-neg 39.0%

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

       8. sqrt-unprod 0.6%

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

       9. add-sqr-sqrt 10.4%

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

   11. Applied egg-rr 10.4%

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

       1. +-commutative 10.4%

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

       2. associate--l+ 10.4%

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

       3. metadata-eval 10.4%

          \[\leadsto z \cdot \frac{x}{t} + \color{blue}{0} \]

       4. +-rgt-identity 10.4%

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

       5. associate-*r/ 10.4%

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

       6. associate-*l/ 23.8%

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

       7. *-commutative 23.8%

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

   13. Simplified 23.8%

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

   if -inf.0 < (/.f64 z t)

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 42.2%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -\infty:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 76.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around inf 73.0%

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

   1. associate-*r/ 78.5%

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

4. Simplified 78.5%

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

5. Final simplification 78.5%

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

Alternative 8: 38.6% accurate, 9.0× 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 99.9%

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

2. Taylor expanded in z around 0 39.0%

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

3. Final simplification 39.0%

   \[\leadsto x \]

Developer target: 97.5% accurate, 0.4× 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 2023274 
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.8867) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) 0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

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